You will never guess what Katie Mitchell is doing next. For 20 years she has been British theatre’s harbinger of doom, the obsessive anatomiser of anxiety, depression, misery and that continental condition known as anomie. While some emerge from her fresh takes on Chekhov frothing at the mouth, loyal audiences go to performances prepared not necessarily to enjoy themselves but to see dark themes played out with matchless precision, often on a dark stage. So her next show arrives from somewhere to the left of left field.

Step this way, boys and girls, for the stage version of Dr Seuss's classic children’s book Cat in the Hat. “When I wrote my application,” she explains, “I said that I absolutely wanted to adhere to the book in a very literal fashion to honour the experience that children have hearing it read.” In practical terms that means staging exact copies of the book’s 30 double-page spreads. “Their expectation is to see the book moving. And I have to say that it’s one of the hardest directing jobs that I will ever do.”

The idea arose from a desire to put on a play that her four-year-old child might like to see. In rehearsal for 50 weeks of the year, Mitchell tries to soften the impact of frequent absences by talking with her daughter about what she’s up to. “This led to her saying to me one day, ‘Well, I told Nathaniel, “Mummy’s doing Dido and Aeneas,” and Nathaniel didn’t say anything back.’ I thought, well, maybe I could find a way of doing children’s theatre. And that would be maybe formally a new challenge, away from bleakness and despair to something very light and different.”

Nicholas Hytner, to whom she answers as an associate at the National, endorsed the idea of a show for children learning to read, and Mitchell duly homed in on The Cat in the Hat. Inspired to encourage infant literacy in 1950s America, it has “the verbal energy that you would need to make live performance”. Mitchell anticipated a struggle for the rights, but the estate read her proposal and agreed. Seuss’s widow is even coming over to see it.

You wonder if they took pains to familiarise themselves with Mitchell’s work. For the past year it has been business as usual, only more so. After Dido was her version of the Purcell opera for the ENO. For the Schauspiel in Cologne she directed Franz Xaver Kroetz’s wordless play Request Programme, in which the lone character puts an end to her life. And now there is a rare revival at the National of Ferdinand Bruckner’s Pains of Youth, which also features a climactic suicide, a symbol of the terrible fate awaiting Middle Europe’s young in the 1920s.

Try explaining that trilogy to a four-year-old. “It’s a bit bleak, even for me,” she concedes of the Bruckner. “I’ve done a lot about suicide recently. Well someone’s got to do it, haven’t they? I’m quite cheery really. I have the emotional stamina to do it.”

Bruckner was the pseudonym of Theodor Tagger, who was born in Vienna in 1891 and emigrated to the US in 1936. His son will travel from America to attend the first night. It marks a break for Mitchell, who will not be teaming up with the video designer Leo Warner. Their collaboration began with Waves, adapted from Virginia Woolf. Their latest work was the Salzburg gig this summer with the Vienna Philharmonic, Luigi Nono’s communist opera Al gran sole carico d’amore. “We probably just needed a break from the scale of what we were doing, just to take stock.” Mitchell also says that she would never use video on a pre-existing play text. “That would be pointless because it is already formally pointing in one direction, so it ought to be executed like that as theatrical realism.”

She came across Pains of Youth — the title of Martin Crimp’s version softens the German krankheit (“sickness”) — in her twenties. “It’s a very heartfelt and pained piece of writing,” she says. “Tagger wants to warn us about what’s going to happen to an entire generation. The play is really a series of scenes that rotate around this awful possibility, which is finally realised.” With this magnetic attraction to self-slaughter, surely her Hamlet can’t be far round the corner?

“No!” she exclaims. “Oh, Shakespeare is so long. It’s beautiful to read, but I really wouldn’t feel I’ve got enough skill to direct it. And I find sections of it parked so much in their period that to communicate them now is almost impossible without having to do funny plane-landing gestures. I’m really foxed by Shakespeare.”

Maybe the clue was there in her decade at the RSC, when she got away with directing only one Shakespeare play, Henry VI, (Part Three), and only then because it reminded her of the Bosnian war. These days Mitchell sees little Shakespeare, or indeed any other theatre: between directing and mothering she gets to about ten productions a year.

It’s perhaps in this isolation, and the freedom to experiment in the National’s rehearsal rooms such as the one in which we meet, that Mitchell has become a lightning rod for audience reaction, above all in her opinion-dividing Seagull.

“I never intended for it to turn out like it has turned out, or for people to have thoughts that they have. You get used to it. It slightly foxes me at times. It is what it is. It is their right to say whatever they want to say, of course, in response to anything they see that you make. And then sometimes I have to hear it and go, ‘Maybe they’re right, maybe there’s a little bit of hubris here so maybe one has to really think carefully about how one does things the next time round.’ ”

I close with a question that I suspect many have wanted to put to her. With the wind in the right direction, would the director of Pains of Youth and The Cat in the Hat consider unleashing herself on something as frothy as a Coward comedy?

“Yeah,” she says, a little startled. “Every genre is interesting. I wouldn’t rule anything out. Also you don’t know who you’re going to be tomorrow, and what shape the environment in which you’re functioning is going to be in. But maybe when I’ve done Dr Seuss, yes, Noël Coward, here I come!”

Pains of Youth runs to Jan 21 at the National Theatre. The Cat in the Hat is at the National Theatre from Dec 11 to Jan 18 and at the Young Vic from Jan 28; nationaltheatre.org

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