Six nights a week for three months – plus matinees twice a week – Gillian Anderson became faded southern belle Blanche DuBois in Tennessee Williams’s A Streetcar Named Desire at the Young Vic, putting herself through three and a half hours of spiralling desperation, alcoholism and abuse.

In one performance she cut her knee open. “I was literally gushing with blood. It was a bit horrific for the whole audience,” she cackles. “I thought I would pass out, which I then proceeded to do the next day about six times.” How did she keep herself sane? “I got a lot of sleep, markedly more than I usually allow myself to. It was very physical, so I saw a chiropractor and physiotherapist on a weekly basis.”

The emotional demands were considerable too. “The first night we performed in front of an audience, I thought, I can’t do any more of this. I felt like a truck had run me over. But by the end there was a sense of catharsis every night – I felt like I got all my anger out, all my tears out, all my whining out, so I felt actually quite peaceful after it.”

The night before we met, Anderson won the Natasha Richardson Award for best actress at the London Evening Standard theatre awards for her performance, beating stage royalty such as Kristin Scott Thomas as Electra and Helen McCrory as Medea. “I’m still completely bowled over,” she says, visibly moved. “I don’t think I’ve ever had a feeling of being an outsider, but it’s definitely nice to be embraced, especially by people whose work I’ve admired for decades.”

The role is one she coveted for 30 years; aged 46, Anderson took on negotiations with the Williams estate, which had concerns that Blanche couldn’t be “too old”. In the end, “absolutely all of the elements came together,” she says. “The Young Vic was the absolute perfect place for it, and Benedict Andrews was my first choice of director. It was a wonderful experience.” Anderson feels she can now bring to the role “a deeper understanding of the grief that surrounds ageing”. She adds that this is “an important part of life, to allow oneself to embrace one’s future without constant self-judgement, and that’s something that never happened for Blanche”.

Anderson has been particularly busy this year. As well as her critically acclaimed turn as Blanche, she appeared in the second season of BBC2 crime drama The Fall and published her first book, a sci-fi novel, A Vision of Fire, co-written with Jeff Rovin, about a child psychologist who has to solve a series of seemingly supernatural events around the world. She has also made her first steps into directing with a short film for the Young Vic, The Departure, a mini-prequel to Streetcar. “I’ve been thinking about directing for a long time. It was always going to be my next step.” She also continues to be involved in various charities, such as campaigning for women’s rights in Afghanistan.

She is modest about these achievements (“it doesn’t even count”) but still, it’s an impressive array of talents. Is there anything she’s really bad at? “Oh, there are so many things that I’m bad at. I’m bad at anything that has to do with IT. I’m bad at remembering anything from my actual life other than lines for, say, a play. Just so many things.”

Anderson was born in Chicago before moving to north London until the age of 11, via 18 months in Puerto Rico. Her family was then uprooted to America, and in 2002 she moved back to London, meaning she has now spent roughly half her life on each side of the Atlantic. “This always felt like home to me, even when I lived in the States,” she says softly in her precise English accent.

Changing schools and continents at the age of 11 was a difficult experience for Anderson, who was teased for her accent and taught herself to lose it (meaning she is now fully bidialectical – able to switch between British and American accents – a useful skill for an actor). A rebellious teenager, she was in therapy by 14, voted by her classmates “most likely to be arrested”, and was indeed arrested on graduation night for trying to glue the locks to her school.

Chris Carter, creator of The X Files, said that when he cast her she looked “like she could have been an Occupy protester.” It’s hard to reconcile this image with the sophisticated woman in 6in stilettos and immaculate make-up ensconced amid the wooden decor and dimmed lights of Fischer’s, a luxurious German restaurant in Marylebone, central London.

Yet, the teenage rebel can be glimpsed. Her hair is tied in a scruffy bun, her fingernails are bare and closely clipped, she is wearing head-to-toe black (“I just scrambled something together”). She says that beauty treatments are “a pain in the arse”, but that playing Stella Gibson in The Fall has taught her to embrace a bit more glamour in everyday life.

“For a good portion of my life I downgraded my femininity. I’m absolutely useless with my own hair and makeup.” There was going to be a scene in the first season where Stella blow-dries her hair, she says, but, “I was burning my neck and couldn’t coordinate it – the whole crew was laughing at me, so it was cut. Now on a semi-regular basis I have my hair blow-dried, and I’m a bit more mindful about what I wear when I’m going out of the house. Just taking care of myself.”

Now in its second year, The Fall sees Detective Superintendent Stella Gibson close in on the serial killer played by Jamie Dornan. It is a role that has drawn comparisons to Helen Mirren’s Jane Tennison in Prime Suspect, though DSI Gibson is much more comfortable with her sexuality. Anderson was drawn to the complexities in Gibson’s character: “Just when you think you’ve understood a little bit about her she’ll do something and you’ll go, ‘Whoa, what are you doing?’ And I like that about her.”

She is curt about suggestions that the show glamorises violence against women (“I’ve been very outspoken about the fact that I don’t agree with that”) and becomes animated when talking about offender conviction rates: “There are still huge holes in a system where women are so afraid to come forward, and still people get cleared. It’s highly possible that if more women were in a position to change the laws, it would be handled differently.”

Previous interviewers’ accounts of Anderson are split: about half say she is witty, playful and warm, the rest use words such as “icy” and “glacial”. She looks slightly bemused by this. “It has to do with all sorts of things – how much sleep I’m getting, if my kids are sick. It’s easy for me to become distracted and protective, and that can come across as cold. At the awards last night my dress felt like it was falling off, so I look very serious in the pictures. I can’t stand there and smile as if I’m having the time of my life if I feel like my boobs might fall out. I don’t hide things very well.”

Today she is recovering from the Evening Standard awards after-party organised by Evgeny Lebedev and Sienna Miller (“four hours of sleep are not enough”), but despite the hectic day she is polite and attentive, pondering each question carefully and frequently breaking into laughter. She is un-chaperoned by a PR, arrives early, and talks for an hour, well over the prescribed 40 minutes.

After the gloom of The Fall and the emotional turmoil of Streetcar, she would like to take on something more light-hearted. “There’s a dearth of comedy anyway, and they don’t generally come my way. I don’t know what I need to do to prove to people that I can be funny.” I ask her to tell a joke and she immediately starts trotting them off. “Why did the hedgehog cross the road? To visit his flatmate.” “What did the zero say to the eight? Nice belt!” She cracks up. “That’s the level of joke I can do.”

So what is she starring in next? Internet rumours that she’ll be in a film called The Curse of the Buxom Strumpet appear to be misguided. “Oh my gosh, is that still on there? Somebody sent me that script years ago, and it had Ian McKellen and Judi Dench attached to it so I said yes.” But she is “in conversations” about a couple of projects, including a New York run of Streetcar and a third series of The Fall.

I remind her that in 1983 she starred in her high-school production of Arsenic and Old Lace as Officer Brophy, a minor role. She roars with laughter: “I hardly remember anything about that at all.” What acting advice would she give herself, back then? “If it was blind instinct that drove me to even audition, my advice would be, ‘Pay attention to your instincts.’”