Picture the scene: in 1979, a young chef returns to London after years of uncompromising training in France, freshly equipped with techniques ranging from classic French to nouvelle cuisine from an eclectic array of placements with some of the country's most exciting chefs, and ready to turn his hand to the kitchen of his family business. That business was Mayfair institution Le Gavroche, and the young chef was the now familiar face of Michel Roux Jr.

Fast-forward 37 years, and the story is similar: Michel's daughter Emily is coming back from five years spent in Paris and elsewhere, having gleaned all she felt she could from more than a few French kitchens. But if you think this is a case of history is repeating itself, you're wrong. "I actually left England when I was 18 to go to a catering college, because I didn't want to have my name here," she explains, on a cold winter day in Le Gavroche's bar, where I'm sitting down with her and her father. "I didn't want anybody to know me."

She describes working at two-Michelin-starred Akrame, under Akrame Benallal: "a really young chef. Really talented, modern – an amazing guy, and a bit whacky sometimes. I came back because I felt it was the right time," she says. "I had been working for the past five years, and got a bit of ground, and been through the different echelons of the kitchen. I just thought it was a good time.

"Walking into the kitchen here when I was 18, having never touched a knife before – I think that would have been horrible." I refuse the urge to mention that I can't believe she hadn't touched a knife, having grown up in the family she has, but I do ask if she felt pressure arriving in France. "No," she explains. "Nobody really knew me over there. In France, Roux is like Smith."

Her father agrees. He would, of course, having done the same thing himself at a similar age. "I think it's important if you are learning your trade not to learn it with your family, but to go out and prove yourself," he says, "to work in different places and challenge yourself, push yourself, which Emily has done. I think that's the best way."

Though their paths into cooking – carving their own mark in the shadow of a famous father – have been similar, this is where they deviate. Michel, of course, joined the kitchen staff at Le Gavroche under his father's direction on his return from France before working his way up and becoming its chef-patron in 1991.

"It's not easy," he says of the alchemy that occurs when working with one's father, when that father has blazed a trail already. "Obviously there's that pressure of working with somebody who is already established – who's got a name and is set in their ways. My style, my interpretation of some of the French classics, was different from my father's, and he would shake a stick at me and say 'That's not how you do it.' I said 'Well, that's how I'm doing it.' It's not easy because you've got to be respectful of the customers, the restaurant and the name. It's almost a burden. It's a big monster to take on."

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But now it's 2016, and it's Emily; not Michel. She may be returning to London, but she's not after a place in the kitchen of this particular two-Michelin-starred restaurant, no matter the trust her training, even before her lineage, allows her. Not permanently, anyway. "I wouldn't work here," she says, assertively, but without malice. "I think it's a huge pressure seeing as it's grandfather and father so, yes, maybe I'm avoiding it because of that. But no, it's not for me."

At this point, you might be lamenting the failure of a long-awaited narrative to materialise: the third generation entering the Gavroche kitchen; perhaps the ceremonial handing-down of a first chef's knife, dripping with allegory. Sorry to disappoint, but it's just not happening. The prodigal daughter, so to speak, will not be spending her days and nights sweating in a kitchen in Upper Brook St.

Instead, something else is happening. Something that feels somehow more suited to now than then; to Emily than Michel. From February, the two will be cooking together in the kitchen at Le Gavroche, but they'll be cooking as equals, on a menu the two have designed together, and that reflects both him and her, under the title The Next Generation. That's right: in 2016, even Le Gavroche is welcoming in a pop-up.

A bit of context: Michel, like many other top-level chefs who've come to realise their restaurants take consume too much of their staff's lives, took the decision – and the financial hit – to close Le Gavroche's doors to the public on Mondays, starting this month. With the weight of two Michelin stars and the decades taken to build an immaculate reputation, it may represent a shift towards restaurants, at the top end and otherwise, giving their staff as much of a weekend as it's possible to have in their industry.

"Closing Monday nights was already in the pipeline," Michel confirms. "We need to address this problem in our industry, because we still work too many hours. It is important that all high-profile chefs take a stance towards this – cutting back hours, better staff retention, better staff working lives – to make our industry more appealing. There is a shortage of quality chefs, so we need to treat our chefs and our staff better, and make it a better place to work."

It's a noble challenge, and one that will surely have ramifications for the restaurant industry across London and beyond. But in addition to that, it represents something else: "I'd been umm-ing and ah-ing about it for more than a year," Michel says, "looking at the figures and seeing if it was viable. It is viable, even if it's a massive financial hit on takings. So that [the work/life balance of his staff] was the driving reason behind closing Mondays.
"But then we said 'Well we've got Mondays closed; Emily is back. Maybe we could showcase some of Emily's work.

"Emily being here, it's just presented itself as being perfect timing. It's going to be so exciting for the customers to see the progression, and to see Emily's style, which is more contemporary and modern."

It's going to be exciting to see Emily's style, which is more modern

"I think it's such a great opportunity," Emily chimes in, "and it will be the first time that it will be my little name on the menu." Although I'd hesitate to call her name "little" (her last name, at least), it's refreshing to hear from someone who could have had everything handed to her.

But she hasn't, and she doesn't seem like one to rest on her laurels: when she's not cooking with her father at The Next Generation series of pop-ups, helping with the family's Chez Roux event offering or finishing the book she's writing alongside her mother, Giselle (off her own back, with no input, and in fact a different publisher, from her father) she'll be very busy indeed: "I'm doing other things," she says. "I'm also working for Restaurant Associates, which keeps me occupied as well."

Michel and Emily Roux in the kitchen at Le Gavroche

Michel and Emily Roux in the kitchen at Le Gavroche

Not only that, she tells me of her plan to open her own restaurant "within five years". I ask whether it'll be a competitor in Mayfair, or somewhere further afield, momentarily forgetting that, having spent five years abroad, she's walking back into a very different London. I ask if it feels that way. "Oh yes," she replies. "And not just in cooking. I came back and I was just, 'Oh God, there's another crane. What's that building? What's this building?' Everything had changed.

"Shoreditch at the time wasn't somewhere you'd think to go, and now it's the whole craze, and there's so much stuff to do and eat there. I still need to explore because I don't think Mayfair would be the best place for me."

Looking shorter-term, though, if Le Gavroche has the clout to influence opinion within the industry in its progressive attitude to working hours, how will this influence extend to the idea of one of the most storied, and in the eyes of some, most traditional, restaurants in London 'popping up'?

"It is an institution and it is iconic," Michel concedes. "It has been here a long, long time and it has got its style. That style means French classic, and it's cosseting, and very comfortable. Some may say old-fashioned.

"There are certain things that I've changed over the years – to lighten up the service, and things like that – and there's more to come. But I have to be respectful of what Le Gavroche is, and Le Gavroche is a place where you can get great French cuisine based on the classics, so I don't want to change anything radically. But you have to evolve – that's important as well. You can't work with blinkers on; you have to open your mind to new practices and new ways. In the same way, as chefs, we open our minds to new ingredients."

I don't want to change radically, but you have to evolve – you can't work with blinkers on

I ask him whether he's mindful of the ripple it could cause – of the painful prospect of this stalwart being perceived to jump on the pop-up bandwagon after so many years of doing things its own way, on its own terms. What if its customers don't want a collaborative supper club? What if they just want a dinner service?

"Le Gavroche has got a certain name for itself," he says. "People see it or perceive it in a certain way, and pop-ups are more akin to being under the arches; a derelict place in the back end of nowhere. I just imagine opening up just for one night, a special night, with different crockery and no tablecloths." He takes a moment to picture it, and makes himself chuckle in doing so.

"That's just it," he says, "'A pop-up at Le Gavroche? Wow!' I think that's fun." ■