I am eating brilliant brains with two of the best chefs in the world. Yielding, delicious, lightly cooked in a pan. "So good," grins Juan Mari, as we wolf them down. No fine dining this, just simple food, spanking fresh: iridescent red mullet and newborn broad beans; tiny peas, potatoes the size of marbles bought from the market only moments before. Elena, his fellow cook and daughter, is immaculate in whites. "They are my most comfortable clothing, like pyjamas," she smiles, as she checks the lunch menu at Arzak, the restaurant which bears the family name.

We eat delicate local anchovies with our fingers, lifting them by the tail. "Fresh anchovies must be eaten by lunchtime," she insists, as we settle to discuss her work, her family, her food.

This year, Elena was named the top female chef in the World's Best 50 Restaurants list, a recognition she says she is proud to accept on behalf of her mother Maite, her aunt Serafina and her grandmother Paquita, who have all helped to make Arzak one of the top 10 places to eat anywhere. The award is also – though she won't say it – recognition of a step out from under the shadow cast by the colossus of Basque cooking that is Juan Mari Arzak.

You can see the excitement and pride in Elena's eyes when locals rush to give her a hug as we stroll around San Sebastián. They all feel involved in the success of this modest 42-year-old mother of two with a stellar palate who helped bring the avant-garde tastes and textures of the Basque region to the attention of the world.

Suppliers shout her name and show off line-caught squid, still flashing like a touch screen and flushing with crimson. One holds a stiff mackerel by the tail and waves it like a sword. Elena points out the sleek sheen on the hake. Not the same, she says, as you get in the UK. "Here, there are deep pools in the sea with special plankton that make it unique. For us, it is the king of fish, subtle, with finesse, and softly flavoured."

While Juan M is mobbed by admirers – it must be like walking around Rome with a popular pope – Elena quietly stands by, chatting with a fish seller about black monkfish and school (Elena has a daughter, Nora, aged seven, and a son, Mateo, five, "always awake, like my father", she laughs). Tenderly, she touches her father's sleeve, whispers a few words, and runs back to buy the live squid for supper. She likes it very finely sliced, dusted with flour and salt, fried till crisp.

"You want a drink?" beams Juan M, as we stop at a pintxos (Basque tapas) stall in the basement of La Bretxa market. Elena has her third coffee of the day, her father his first glass – "doctor's orders, only red wine and champagne," he says. We eat bacalao with perfect slippery potato while he explains how to spot hot peppers from sweet. It is all about the white hairs, he says. He can also tell whether a dish has been cooked by a man or a woman, he adds, though here I am less convinced.

Juan M is nearing 70 now and hasn't slept in two days, he says, having flown in from Mexico where the family once owned a restaurant. But while Elena and I try to keep up, like children being dragged on a leash by a large dog, he strides ahead to the old town for a plate of his favourite prawns. "You want a drink?" he asks, proudly pointing to the "San Sebastián-style London gin" he is producing with his friends behind the El Tamboril bar. The prawns – gambas a la gabardina [in a raincoat] – come wrapped in an ethereal batter. Shopping finished, we take a car to the restaurant to talk and eat, until suddenly Juan M calls a stop for chorizo. "You want a drink?" he smiles, eyes sparkling, almost innocently.

Elena and her father's lives mirror each other in many ways. They both discovered their love for cooking in their school holidays. They both studied with the great chefs of their day – Juan M in France with Paul Bocuse, the Troisgros brothers and Alain Senderens; Elena with Alain Ducasse, Ferran Adrìa and Pierre Gagnaire. But, more importantly, they've both been shaped by cooking in the San Sebastián restaurant their family has owned for 115 years.

Paquita took over the kitchen at Arzak on the death of her husband when Juan M was nine. "He was an only child surrounded by women, in a matriarchy," says Elena. "I think that is why he idolises women now."

"The restaurant was different then," he adds. "Our bedroom opened on to the dining room, there was my mother, my grandmother among five cooks, not the 30 now, serving traditional food. But I was very happy."

By the time Elena was chopping vegetables as a child, Juan M was maybe the most important chef in Spain, famed for the revolutionary nueva cocina vasca he helped create with fellow San Sebastián chef Pedro Subijana.

Like her father before her, she cherishes her time as a child in the kitchen. "We were only allowed to work for two hours a day in the holidays," she says. "I always wanted to stay longer. I adored being there with my sister, my mother, my grandmother, my aunt, my family.

"My mother was in my father's shadow," she says softly. "It was harder for women then, but I trust her palate so much. I owe her everything."

Elena was always an adventurous eater. "I can remember the smell of my first truffle when I was eight, and a txangurro [crab] dish, without laurel [bay] as we like it here. My parents also brought home wonderful cheese from their travels."

Good at school, she was, though, always determined to be a chef. Calming her concerned parents by first passing her university entrance, she studied and cooked outside Spain – including a six-month stint at Le Gavroche in London, aged 20 – before returning to San Sebastián.

"I spent six years abroad," she says. "I had a cocktail of ideas, but I had to find my identity at Arzak, otherwise I wouldn't be useful. I worked in each station of the kitchen to relearn what happens in my house. I also studied many books of Basque cooking to learn more about my roots."

She discovered her style in San Sebastián's markets. "In the 90s, chefs here were cooking with mangoes," she shudders. "I saw the local produce, the small peppers, other things, and I thought why are we not using more of this in modern food. After four years I had found my voice."

Introducing her ideas for a cutting-edge Basque cuisine took time, patience and intense negotiation. "My father risked very much with the ideas he started to make with me," she says. "Sometimes he would say, 'the sauce is horrible, we will change it' and we would lose half the plate. I think my father thought, 'If I don't allow Elena to make her new things she will get bored and leave', but he also knew that if I introduced a tandoor, ginger, a mole from Mexico, it would work with the tastes of my region."

A world-beating father and daughter team was born. "Sometimes I wonder whether my father would have evolved in another way," she says. "Perhaps he would be better. But we have been working together for 17 years. I am 42 now." Even today, both must agree on each new dish that appears on the menu (although Elena is still smarting over a sea urchin gratiné her father vetoed 15 years ago).

Back in the kitchen at Arzak. I ask Juan M what makes his daughter one of the best chefs in the world. He pauses, smiles and orders vintage Cristal to accompany our simple bowl of potato and peas. "Primero, she is very ordered," he says quietly. "Segundo, most important, is her palate. To be a good chef, you must know what is well done and not. Even where we are now with our cooking, we recognise Basque tradition."

She has always had the sensibility of a chef, he says. "When she was five years old I took her to the best restaurant in San Sebastián. She asked for her meat 'less well-done but warm'. I was embarrassed. Even as a child, she was always showing others what to do."

It is Elena's turn to be embarrassed now. "What we eat, how we eat, is in our culture," she says, taking the conversation into a more comfortable area. "Our signature cuisine is Basque. My father and I talk about this all the time. Our taste is from here. We were born here. Since childhood we have in our minds a flavour: parsley, squid, hake throats. Even if we use herrings or dill, we cook unconsciously with this identity."

While Juan M pours more champagne, Elena tells me that he cried for two days when told of her award.

"I am 70 in a month," he shrugs. "My life is very long. It has been very special. She is a woman, I am a man. But in cooking there is only a plate. Sometimes she makes dishes I don't like. Sometimes I will make dishes that she won't like. But we make many, many dishes on which we agree.

"I have given a lot to her, she has given a lot to me. Our cooking is not from Juan Mari. It is not from Elena. It is Arzak!"

After seven extraordinary, exhausting hours together, Elena sums it up. "I am Elena Arzak," she says simply, "the daughter of Juan Mari. The possibility of cooking was always in my heart."

For more information about the food and culture of San Sebastián, see sansebastianturismo.com