Hint: It's deeply felt.
"I had a type of (being) starstruck the first day on the set that I've never experienced before," Burrell says. "It's a funny thing to be so starstruck, but I was genuinely intimidated.
"My first scene was in the train station, where I was saying to the Muppets, 'Come with me for the investigation.' And it was all of them. And I was so nervous. I was so nervous, it took me five takes, and I have, like, one line."
Fey says of co-starring with Kermit the Frog, "He's the biggest star I will ever work with, certainly. You see him in a press conference, and people are just psyched that he's there.
"You just have to play it cool. You have to act like you belong there with Kermit. Fake it till you make it."
In "Muppets Most Wanted," opening Friday in Houston, the whole gang is back for a sold-out world tour. Sounds peachy, but it turns out they're being hornswoggled by their new manager - who would have guessed that Dominic Badguy ("Bah-gee," he explains it is pronounced) would have ulterior motives? Although he is played by Ricky Gervais.
Badguy is in league with the world's most dangerous frog, Constantine, who happens to be a dead ringer for Kermit. When a series of audacious robberies follows the Muppets on their European swing, hardboiled Yankee law bird Sam Eagle teams up with an intensely French Interpol agent (Burrell) to solve the mystery. Meanwhile, Kermit finds himself in a Russian gulag that is ruled with an iron fist by a Broadway-loving guard (Fey).
"All of it was a joy. The hardest thing was trying to stay Russian and not turn Italian."
Fey says her accent was based in part on villainous "Rocky and Bullwinkle" femme Natasha Fatale:
"I studied Natasha, yes, and I worked with Jill McCullough, our dialect coach, and then I just tried to channel every woman who has ever given me a bikini wax."
The eight-time Emmy winner and youngest-ever recipient of the Mark Twain Prize also admits her dialect work might have worn on her family at times: "I imagine they got quite tired of it. It still pops up every now and then. Sometimes I'll be changing a diaper and it's like," she drops a few tones and raises an eyebrow, "You give me poop. I put poop in trash can."
"Outside of Clouseau, it was probably every single French character on film - and impressions of French people. That's how broad the accent is. Luckily for me, James was like, 'This is not about subtlety. We're doing a collection of all the French inspectors ever.'
"That's why it's so fun being on set - there's no real ceiling for whatever it is. The Muppet world can hold whatever combination of being smart and ridiculously silly you can come up with; you just don't feel inhibited."
Fey, like Burrell, a parent of two young daughters, grows slightly sheepish when asked if her kids were a prime mover in her taking the role: "I am excited that they can watch it, and I'm excited that they're very excited for the movie to come out so they can see it. But I have to be honest that it's really - this is fulfilling my own Muppet fandom. Whether they had been into it or not, I was going to do this."
The Second City vet praises the off-the-cuff chops of the Muppeteers.
"They're great improvisers, they really are legit, yeah. The shots and everything are really kind of complicated, so it's not like you come in improvising the whole scene. But if you ad-lib within a scene, they're right there with you."
In publicity quotes, Kermit has claimed Fey was so "method" that she would leave him locked up at the end of shooting days.
"I would only speak to him in character," she acknowledges. "Also, we refrigerated the whole set so it would feel like Siberia. But I think you see onscreen that it was worth it."
The actress says she did have minor conflicts with Miss Piggy over spending so much time with Piggy's amphibian amour.
"I did. But you know, I think she knew it was just a role, and she knew she is physically stronger than me, and if anything (happened), that would solve it."
Burrell says his intimidation came more from awe:
"When you run into celebrities you admire, their humanness is much more apparent than it ever was onscreen. People are often not wearing any makeup, you know - there's a lot of movie magic happening - you see people's age more. There's something about it that actually I find disarming, in a wonderful way: 'Oh right, they're human.'
"But the Muppets look exactly the same. Identical! So I was like," he guffaws, " 'Oh my God! You are kind of like gods in a way because you're not going to age. You're exactly what I remember.' "