The cuisine - Hampshire trout, roasted Swaledale lamb, British seafood - will have to catch up, too: it is patriotic but old-fashioned. Balshaw reaches for the menu - big rings on each hand flash across the table - and asks: "Would you like one course or two?" We agree on two because "we open Fahrelnissa Zeid [at Tate Modern] tonight so I won't get home to have my tea till 9pm," Balshaw explains, scrutinising the starters. "It's a toss-up between asparagus or Cornish crab: I'll go for the crab in honour of St Ives!" She chooses pearl barley risotto as a main; I opt for British asparagus followed by Gressingham duck.
If the crab nods to Tate's regional galleries in Cornwall and Liverpool, the word "tea" is also a giveaway. Not only is Balshaw Tate's first female director; she is Midlands-born, into a "family of public servants, social workers, community workers, teachers, that sort of thing", state-school and non-Oxbridge-educated, and forged her career in the north of England.
As director of Manchester Art Gallery and the university's Whitworth Art Gallery, which she transformed into a state of the art contemporary venue, she made her name globally as Manchester's unofficial cultural attaché and an intellectual heavyweight of the northern powerhouse.
"What holds the country back is if you have a sense that what's happening in London is the only thing that counts," she says. "London is exceptional but not everyone can afford to live there. It's vital that other cities are vibrant. In Manchester or Liverpool you can take risks, you're not constrained by the tourist audience. London can learn from the regions."
At a time when the cultural divide between metropolitan privilege and the rest of the country feels bitter, Balshaw's appointment is a vote against elitism, and for both unity and change.
"I went to an unprepossessing comprehensive in Northamptonshire," she says, "and one of my first letters of congratulation was from Mr Cartwright the drama teacher, who wrote, 'I'm so proud of you - and I remember that your socks always matched!' I was rebellious but I didn't want to break all the rules. The only items of uniform not prescribed were socks and gloves, so I wore dayglo or bright orange ones - I wanted to not conform, but not get into trouble. Mr Cartwright also said it was 'not very usual for someone from a school like that to end up director of Tate'. Hence my vision for any child coming to Tate is that we are open to them in whatever way they need."
Balshaw's career has been full of nonconformist moments, none getting her into trouble - from serving takeaway curry at the Whitworth's inaugural dinner to rethinking global collaboration in her New North South initiative this year, linking northern English arts institutions with South Asia. And she tweets on everything: her Christmas gold brogues, her cracked wrist ("from a big man falling on me in yoga class"), Nigel Farage.
Her drive, she says, is for art to connect with the widest range of people, but she admits, "We have a long way to go to reach people who might not think any of the art is for them."
Can a wider reach sit comfortably with high art?
"It's not losing the artistic adventure to speak to a whole spectrum of people. Our museums have better-educated audiences, more white than non-white, which don't reflect their cities' demographic. We can afford more ambitious targets. It's less about what you show, more about how you connect. We used to think museums were irrelevant to 16-to-26-year-olds, but you have to be open at different times of day - then they come. At last Friday's LGBTQ Late at Tate, they were there in their thousands: they effectively took ownership of that time and space. Tate at 10pm on Friday night feels very different from 10am on weekday mornings."
Balshaw salutes Tate Britain's current Queer British Art show - "telling incredibly compelling new stories about the art which people think they know", curated with help from LGBTQ groups - as a model of "different ways for institutions to work with diverse communities". I thought the show flawed - social commentary at the expense of significant work. As we begin our starters - brown and white crab with a poached peach and almond mousse, asparagus with crisp polenta and softly boiled duck egg, both pronounced delicious - I overhear the architect and his wife express my own old liberal reservations. "Art shouldn't be defined by an artist's sexuality, should it? Unless it's prurient - and then I don't want to see it!" But they decide to give it a go: "Homosexual art, here we come!" they cheer, and the wheelchair sets off.
If Balshaw can even energise the over-90s, she is winning. "I want Tate to ask: what is the history that we want to tell from now?" she says, arguing that cultural studies, now standard in academia, have had a profound influence on artistic practice. She points to museums and galleries that have a firm grasp of this fundamental shift - Nottingham Contemporary's "terrific" show of black British art in the 1980s, for example. "Young people today have a sophisticated grasp of identity, its fluidity, how it is culturally defined," she says, adding that her own children - her son and daughter with her first husband are young adults; she married Manchester Museum director Nick Merriman in 2010 - "don't understand it as post-structuralist theory but as lived reality, and that's tremendous."
Balshaw's career began in academia. She has a degree in English from Liverpool and a doctorate in African-American art and literature from Sussex, where "queer studies, post-colonial studies, were on fire in the 1990s, Shakespeare scholars were teaching sexual dissidence". Her lightning moment in visual art was seeing Cornelia Parker's "Cold Dark Matter" at Chisenhale Gallery in 1991: "I walked in and there was an exploded shed, it's pretty memorable. Until then most things I'd seen had been in a frame. I appreciated the daring, the sense of jeopardy, which gets to something I'm drawn to: the pull of terror and joy."
A quarter of a century later she opened the new Whitworth with Parker's piece: "I kept having to pinch myself, I remembered being 21, then I was 46." There have been other completed circles. Tate Liverpool opened in 1988, the year she became a student, and was the first gallery she came to know - "I remember a Richard Long piece, feeling comfortable enough that I could sit on the floor and just be with it".
Being steeped in this background distinguishes Balshaw from her predecessors, traditionally-trained art historians Nicholas Serota and before that Alan Bowness (both Cambridge then the Courtauld). Serota embraced conceptual art, building a whole new museum, the Switch House, to display it. Balshaw, a quarter of a century younger, brings the concerns of the next generation, a desire to understand how history has been written and shaped. "It's not that there weren't women artists in the 1920s or 1950s, it's just that history hasn't focused on them."
Is she a feminist?
"How can you not be a feminist today? I'm often asked of an artist, 'Do you show her because she's a woman?' - the answer is, 'No, we show her because she's really brilliant.' I don't strive for shows 50 per cent by women because that's a feminist gesture, but because we want to refer to the world we live in."