It can’t be easy being cast alongside Mark Rylance at the moment. The former Artistic Director of Shakespeare’s Globe won just about every theatre accolade going in the past year for his remarkable performance in Jez Butterworth’s hugely acclaimed play Jerusalem. In Matthew Warchus’s revival of the Laurence Olivier Award-winning comedy La Bête, the talented performer is given the space and time to deliver a 30-minute monologue packed with tricks and party pieces while the rest of the cast, which includes 11-time Emmy nominee and Tony Award-winner David Hyde Pierce, watch on as he steals the show. Surely for a performer of Pierce’s stature, that must grate just a little? Apparently not.

“His gifts are unbelievable as an actor and there are certainly actors with that kind of talent – especially playing the kind of role he’s playing, which is a kind of obsessive, full-of-himself vulgarian – who would be really good at that part but you would not want to be on stage with them. Mark is not that man. He is an incredibly generous, gracious actor, which is not to say he’s not challenging on stage. He loves to play and I love to go wherever he ends up going.”

Pierce, who is chatting to me on the phone from the comfort of the hotel room that has become his London home during the run of La Bête, is enjoying his time in the capital. While he may be missing his dogs, the feeling of settlement that comes with actually living in London, rather than being a tourist, is bringing out the best in the actor best known for playing younger brother Niles in long-running US sitcom Frasier.

“I think our show takes people on a trip that they’re just not expecting,” he says of the David Hirson comedy about the clash between a high-minded classical dramatist (Pierce) and a lowbrow street clown (Rylance). “I don’t think, even if you had read the play, you would expect where this particular production takes you.”

This inventiveness in the revival of the show that won the Laurence Olivier Award for Best Comedy in 1992, he puts down to director Matthew Warchus, for whom he has as much admiration as he does for Rylance: “I am amazed at his ability to really find all the elements of this play; its darkest moments, its funniest moments, the broadest farce, the smallest heartbreaking things. To mine all of these things and have them be present in the production, I’m in awe of him for that.”

It was the combination of Warchus, Rylance and the play that originally attracted Pierce to the production when American producer Scott Landis called him with the proposition. Having signed up following a read through of the play in London, the casting of Joanna Lumley, in a role that was altered for this production from a prince to a princess to include “a strong woman’s voice” in the play, convinced him he had made the correct decision. “It told me, without ever having worked with Matthew, that we were on the same page as far as how we viewed the play and what we imagined it to be.”

Pierce’s involvement with La Bête could have been very different. He auditioned for the original Broadway production, which ran at the Eugene O’Neill Theater in 1991. He didn’t get the role, but swiftly moved on to be cast in Frasier. La Bête ran for just 25 performances, closing after a stinking review from New York Times critic Frank Rich. Though Rich may not have liked the piece, “Every single person I have spoken to who actually saw the production,” Pierce tells me, “has said ‘That was one of the great theatrical events of my lifetime. I’ll never forget it.’” It also received five Tony Award nominations.

Pierce is aware, then, of the risky nature of this revival, which runs in London before a Broadway transfer that was booked before anyone knew how popular the piece would be. There are many expectations flitting around the production: the Olivier Award and Tony nominations, the public feeling, the aforementioned damning review. The show’s history leaves it open to more comparisons than if it were on a website advertised by a meerkat. This is why, along with Rylance and Warchus, the show’s producers, which include Sonia Friedman, receive so much of Pierce’s praise: “I am thrilled that these producers are willing to take a risk on something that is not the most obvious choice to do in tough economic times. The choices that they’ve made, as far as I can tell, have all been artistic choices as opposed to financial choices. They chose to cast an entire company – half Brits, half Americans – that will go with the play. The producers’ hearts are in the right place, but they’re not blind either, they know what a risk this is but they’re up to the challenge.”

Pierce knows all about challenge, his natural tendency being to push forward with a new goal. After 11 years playing the lovelorn, neurotic Niles in Frasier, collecting an Emmy nomination in each year of the show’s run, it would have been easy to have become a sitcom regular. Instead, he set his sights on trying his hand at a slightly different field, musical theatre.

As a child he had played the piano, and had even gone on to study classical piano at Yale University, so music had always played a large part in his life. But, as he says, “You don’t just walk in and do eight shows a week of a Broadway musical. You need stamina and vocal training and all these other things.” So, during his time on Frasier, he invested in both vocal coaching and dance lessons so that when the show came to an end he would be ready for that next challenge.

The challenge in question just happened to be Monty Python’s Spamalot, in which he originated the role of Sir Robin. It was another production that came with added pressure. The Pythons’ comedy is loved by many; any unsuccessful messing with a treasure like Monty Python And The Holy Grail, on which the musical was based, could have been greeted with taunts more pointed than ‘Your mother was a hamster and your father smelled of elderberries’. Pierce’s nerves were settled after the first read through of the script, at which he performed all of the Eric Idle parts while perched next to the former Python and creator of the show. “If I could do Eric Idle in front of Eric Idle, then I didn’t worry about what anyone else thought.”

From Spamalot, Pierce went on to star in the Kander and Ebb musical Curtains, for which he won a Tony Award. When I ask about the coveted accolade, the highest you can get in American theatre, he sounds distinctly underwhelmed. “It was a very nice thing,” he says.

I guess when you receive an award nomination every year for 11 years the novelty of recognition wears off. So what of the time he spent in what is universally accepted as one of the greatest sitcoms of all time?

“It was 11 years of, really, a great group of actors, astonishing writing and a lot of shared life. That’s a long time, so we all have those things in common; cast members getting married and having children. Several of us had our parents appear in guest slots just sitting at the bar or in the coffee shop, and now some of those folks are gone. There’s a lot of deep, rich feeling associated with that show.”

I get the feeling, though, that unlike so many successful TV series, Frasier won’t be receiving a television revamp, movie or musical makeover any time soon, certainly not with the involvement of Pierce. He says, of television in general, “I don’t have a lot left undone that makes me want to go back,” so why would he return to a rehashed version of the sitcom? Better to let it live on in ageless reruns.

He has a new project fermenting, which may or may not come to fruition following the end of his journey with La Bête. Like so many actors, he won’t give too much away this early on, as nothing has been confirmed, but I can be fairly certain it won’t be a stint on a reality TV show. “It does seem to appeal to some of our lower instincts about enjoying people being humiliated and being able to feel better about ourselves because we see someone who isn’t talented trying to do something,” he says of the genre that rose in popularity while he was starring in Frasier and now dominates the airwaves.

I venture that this battle between the lowbrow and highbrow that is being fought in television scheduling meetings around the world, is reflected in La Bête. “As much as it’s about high art and low art,” he replies, “it’s about ideals versus real life. We all have those two things in us; our better selves that we wish we could be all the time and the reality of daily life. The play’s very much about what it’s like to be an insider and what it’s like to be on the outside; Mark’s character being completely outside the world of the play, trying to get in. It ends up being a debate and a contest and a question of who will stay on the inside – the in group – and who will be cast out … like a reality show.”

Stillness is a great quality in an actor, and a rare one. David Hyde Pierce is pretty much still most of the time. We are sitting in his underground dressing room at the Comedy Theatre, where he has just opened, to very favourable reviews, alongside Mark Rylance and Joanna Lumley in La Bête, a rhyming play about 17th-century French actors.

I say that I won’t even mention Frasier. He neither twitches nor smiles. Then, of course, I do mention Frasier. Same reaction. He’s placed his early lunchtime tuna baguette to one side – there’s a matinée in two hours’ time – and offered me a glass of apple juice. He’s concentrated, watchful, Buddha-like, even though he’s wearing sportswear and trainers. His crash-pad day bed is tidily arranged in the corner.

I realise now that those slow burns and sudden eruptions of mania as Dr Niles Crane in television’s most popular ever sitcom – it ran for 11 years between 1993 and 2004 – are rooted in a steadily controlled lifestyle and demeanour. If Pierce has high blood pressure, I’m a Dutchman.

There’s not much small talk, not with journalists, anyway. He was born and raised in Saratoga Springs in upstate New York, and went to Yale to study music, switching to a double masters in English and theatre studies. As a student he played in Beckett and Gilbert and Sullivan, and made his Broadway debut, auspiciously enough, in Beyond Therapy by Christopher Durang in 1982.

Was it true that the role of Niles Crane, Frasier’s younger brother, was created for him because he looked like Kelsey Grammer? “In a word, yes. The casting director said to the writers that if Frasier were to have a younger brother … and so they came up with me and the construct of two brothers, two therapists with different kinds of therapy and different educational backgrounds. And that was that.”

Wasn’t it a grind, year in, year out? “Not at all. With a live audience very week, and those scripts, it was always perfect for me. And there is always time off; I’d have four months out and do a movie or a play. One of the great experiences of my life was doing a two-hander, Six Dance Lessons in Six Weeks, onstage with Uta Hagen; it was the last thing she ever did, and there was something magical in our partnership. I learned so much from her, just being with her on stage.”

Talking of legends, Pierce also worked, pre-Frasier, with Peter Brook, playing the house servant Yasha in a specially re-vamped US version of Brook’s Bouffes du Nord Cherry Orchard, alongside Brian Dennehy, Linda Hunt and the great Ingmar Bergman actor, Erland Josephson. So his renewed stage career is a happy return, not a tentative new start.

We all know he is, as the New York Times put it in 2000, “a comic-pathetic actor of genius, a sort of modern day Buster Keaton for TV.” So had he read the reviews, which have been mixed, of La Bête? He gives me a look which says, “Why would I have done that?”

The trouble with his role, that of Elomire (or Molière the playwright), is that he has to stand and listen to Rylance as the upstart thespian Valère spouting brilliantly through false teeth for half an hour. I was going to ask if he felt that acting was as much about listening as speaking, but the question sticks in my throat, already unworthy of his consideration.

“When my sister read it,” he offers – at just turned 51, he’s the youngest of four in a close family, many of them recently over for the première – “she said, what do you do all that time?” One critic has suggested that he’s slowly realising that he’s playing the wrong part. I’m not even going there … “In a way, we are the audience on the stage. Everyone knows someone in their life, hopefully not as extreme as Valère, who prattles on so you can’t actually believe they’re still talking.”

So why had he taken the sponge-like role of Elomire? “The project came to me, although the writer, David Hirson, reminded me – I had completely forgotten – that I’d auditioned for the role of the Prince [it's now a princess, played by Lumley] in the first production in 1991. And not got it. It was near the end of my time in New York, before I went out to LA for Frasier, and they hired a fine actor called Dylan Baker. I’m told I was very close.

“I never saw the show, but it’s remembered for two reviews: Frank Rich’s in The New York Times, which killed it; and John Simon’s in New York magazine, equally dismissive, which was written in rhyming couplets. When I re-read the play, I wasn’t interested in the Prince, but Elomire fascinated me. Our director, Matthew Warchus, knew there was a dark side to the play, and we discovered that; it was always going to be worth the journey.”

But is the piece really any more than an over-contrived conflict between the meretricious and the serious in theatre? “Matthew describes it as a conflict between idealism and reality, and each of us has our better self and our practical self, and every day is a process of juggling those claims on us. And there’s also what it means to be an insider and be cast out, which is what happens to Elomire. He’s an insider in a world Valère wants to gatecrash, and it may be poisoning him … “

When he first left Frasier, Pierce bounced on to the Broadway stage in the Monty Python musical Spamalot, playing Sir Robin (“not quite so brave as Sir Lancelot”) and leading a flat-out funny ensemble number about the need for Jews in all Broadway shows: “There’s a very small percentile who enjoy a dancing Gentile.” I saw the show on its first weekend, with queues of obese Mid-Westerners, savvy New Yorkers, musical theatre buffs and geeky Python fans snaking right round the block and down to Times Square.

“I’d been waiting for this for many years, and had been in training, and singing in benefits. I knew that once Frasier finished, musical theatre was the next thing for me. I wasn’t obsessed with Monty Python but, when I was a kid, after I’d finished my organ practice one night in the church, the show came on a local PBS channel; it was the sketch about the tedious life of a chartered accountant, not even one of their best sketches, but it was bored into my brain from that moment.”

The fact that the Spamalot director was Mike Nichols hit another button: Pierce was a devotee of the deadpan cabaret sketches Nichols recorded with his comedy partner (and then wife) Elaine May. “Mike and Elaine’s work was a huge influence, so was Bob Newhart’s … and Alec Guinness. They all made a big impression, but there was also something in me drawn to their sardonic style of humour. But I enjoy slapstick, too. In Frasier we had the chance to do both.”

I think of Niles with a parakeet on his head at a dinner party, or setting his pants on fire, or pretending that a sack of flour is his baby, or trailing around after Daphne Moon like a spaniel with indigestion. Are the Frasier gang still close? “Totally. We went through a lot together, in our personal lives as well. I’ve just been to see Kelsey in La Cage aux Folles on Broadway, and I came to London for Jane Leeves’s wedding.”

Ah, so there might be a stage version, perhaps even Frasier: the Musical? Another blank look. “No, we don’t want to do anything like that. It’s something we’re all very proud of and are happy to leave in syndication. Life goes on.” And he loves London. Is he bugged when he goes round the shops? “No. People are friendly when they recognise me. It’s a great gift to be able to pretend to live here for four months. It feels like home.”

He’s been to Evensong in St Paul’s and the National Gallery and is planning to catch as many concerts as possible. His last Broadway show, for which he won a Tony award, was a “theatrical” musical, Curtains, by John Kander and Fred Ebb, and he went the other night to see a Guildhall School production of it. “It was very moving. I cried, seeing this show we had created done by these amazing kids, with a really fine orchestra.”

His partner, TV writer and producer Brian Hargrove, has just returned to New York, where La Bête follows in September. I wish him and the play better luck this time round with The New York Times. He doesn’t respond, but shows me with an almost eerie politeness to the stage door. And he shuffles quietly back towards his bunker and his baguette.