Elisabeth Moss, 37, has been nicknamed “the Queen of Peak TV” for her award-winning roles in several of the 21st century’s most acclaimed dramas. She played the president’s daughter Zoey Bartlet in The West Wing, secretary turned copywriter Peggy Olson in Mad Men and detective Robin Griffin in Top of the Lake. She currently plays June “Offred” Osborne in The Handmaid’s Tale, which she also co-produces. She stars in the new film version of HG Wells’s The Invisible Man.

The Invisible Man seems a bit of a departure for you. What attracted you to the role?

I’ve always loved horror films and secretly wanted to be that girl running around screaming. This role not only offered that but was an interesting character piece with deeper resonances.

The story has been reimagined so it’s about a woman trying to escape a coercive, controlling relationship. Did that feel timely to you?

Absolutely. [Writer and director] Leigh Whannell’s immediate take was: “What if you told it from the perspective of the victim?” because that had never been done. I thought it was so brilliant and totally timely for this era.

Can genre films like this and Jordan Peele’s Us, which you also appeared in, tackle sociopolitical issues by stealth?

That’s what the best horror films do, like The Shining or The Exorcist. There’s far more to them than what’s on the surface. The fantastical element means they can address those things without feeling like a moral lecture. It’s the same in sci-fi – Gravity isn’t just about astronauts but grief and the loss of a child. Genre films are at their best when they’re a lens through which to view issues.

You’ve said The Invisible Man is one of your most challenging roles. Is that because it was so physical, with fight and chase scenes?

People have got used to seeing me in emotionally challenging stuff, so they assume it’s the physical side that’s harder for me. And they’re exactly right! But I did as many of my own stunts as possible and really enjoyed it.

Your character has a doberman dog called Zeus. What was he like to work with?

A total diva! He’d never come out of his trailer and was very demanding about the catering. No, he was so sweet. They say never work with animals and children but I’m happily ignoring that rule.

What would you do if you were invisible for a day?

Go to a film set or theatre and watch an actor up-close who I really admire. Someone like Meryl Streep. I hope she doesn’t read this and take out a restraining order! That’s the least subversive thing I’d do, anyway. Everything else would involve sketchy, shady stuff.

You move between film and TV. Which is your natural home?

Luckily, I don’t have to choose any more. I can make a film that gets a short release in cinemas, then moves onto a streaming service and finds a whole new audience. People who were traditionally TV actors are doing films, while Meryl Streep is doing TV. Oh, I mentioned her again. That restraining order just got tighter!

You’ve starred in several of the best TV dramas of all time. How do you choose your parts?

It’s partly luck, partly instinctive. I look at the script, first and foremost, because writing is ultimately the only thing that really matters. If I find myself saying the lines in my head, that’s always a good sign.

You start shooting season four of The Handmaid’s Tale next week. The role is pretty gruelling at times. Does it take a toll on your mental health?

It’s been a great place for me to exorcise my emotions, honestly. Playing June has inspired me to be a stronger woman and advocate for issues I think are important. It hasn’t taken a toll, it’s helped me – in the same way that I hope it’s helped our viewers.

The costume has become a form of feminist protest. How does that feel?

It’s become a symbol of something so much bigger than a TV show. I feel proud when I put it on. June might wear a cape sometimes but she’s not a superhero. She doesn’t have any special skills except following her heart and refusing to give up.

One of my favourite scenes was when June finds a DVD of Friends…

Oh my God, how great was that? Any time we can incorporate reality into the world of Gilead, it’s an important reminder this is not an insane dystopian future, this is speculative fiction. It’s our world, our time, with parallels to what’s happening now. We’re doing more of that in the new season, actually.

Elisabeth Moss as June in The Handmaid’s Tale. Photograph: Barbara Nitke/Hulu

What will June be watching this time? Frasier? The Big Bang Theory?

[Laughs] Maybe she’ll be watching Fleabag.

What did you make of Margaret Atwood’s sequel, The Testaments?

I’m reading it at the moment – except I keep getting bits spoiled for me by people on the show! It’s an unusual situation, like Game of Thrones, to be adapting a story that’s still being written by the author.

You’re in Wes Anderson’s next film, The French Dispatch. How was being in his world?

A surreal, dreamy experience. Just going on set, I felt like I was stepping into a Wes Anderson movie. I was dressed like I was in a Wes Anderson movie. All the people who appear in his movies were there. Even Wes Anderson looked like he was in a Wes Anderson movie.

Who’s on your bucket list to work with?

I’d always wanted to work with Wes Anderson, Scorsese and the Coen brothers. I’ve just crossed off Wes but I’d love to collaborate with him again. And there are directors coming up who I admire, like Jennifer Kent [The Babadook] and Alma Har’el [Honey Boy].

You’ve won two Emmys and two Golden Globes. Where do you keep them?

Three Golden Globes, actually. But who’s counting? Just kidding. They’re on a bookshelf in my apartment, along with a bunch of memorabilia and trinkets from other things I’ve done. I should probably get something nicer like a glass cabinet.

The Invisible Man is in cinemas from 28 February