Juan Mari Arzak led Spain's culinary revolution and became a King of Nouvelle Cuisine. Soon his extraordinary daughter will inherit the stove.

At restaurante Arzak, in the Basque city of San Sebastián, I am contemplating the perplexing black food on my plate when a waitress approaches me with a pitcher. As she pours a stream of warm broth onto the dish, it slowly melts away all the darkness to reveal a cluster of blindingly bright orange discs. It's like watching the sun come out. The dish--pumpkin ravioli veiled in squid ink--is a postmodern showpiece, as stylish and startling as it is delicious. "Serious but entertaining--that's the food I'm trying to do," explains Elena Arzak, 32, who is quietly coming into her own in her father's legendary kitchen.

For 10 years now I've been returning to Arzak, where Elena's larger-than-life dad, Juan Mari Arzak, led Spain's nouvelle cuisine revolution more than two decades ago and gave the country one of its first Michelin three-star restaurants. Everything I've ever eaten there has been remarkable, but something magical has been happening lately. Like crisp smoked potatoes fashioned into Yamamoto-style pleats with sizzling prawns concealed in the folds. Like the Basque breakfast--spicy chistorra sausage and egg--deconstructed into a mesmerizing geometric abstraction. As I taste a smoked chocolate mousse, it occurs to me that Elena Arzak is the most exciting woman chef on the planet.

I announce as much to Elena. Big mistake. She shakes her head furiously (an anti-diva) and implores me not to publish such nonsense. And yet ... The two other female chefs presently cooking in Michelin three-star kitchens--Nadia Santini, of the hypertraditional Dal Pescatore, and Luisa Valazza, of the rather staid Al Sorriso, both in Italy--can hardly compete with Elena's futuristic techniques. Britain's restaurant scene remains testosterone-driven, and there are currently no women in America mentioned in the same breath as, say, Thomas Keller or Charlie Trotter (Alice Waters being a force rather than a chef). And France? Gilles Pudlowski, one of the country's most ubiquitous and influential food critics, insists that French female toques (like Reine Sammut, of La Fenière) would give Elena Arzak a run for her truffles; but even he concedes that "if in the past, Spanish chefs traveled to France for fresh ideas, now it's our turn to cross the Pyrenees."

Born into Spain's premier food family and raised in a city that enjoys mythical status in the haute cuisine world, Elena earned her first pocket money helping out in the kitchen. ("I was the one doing the dirty work, like cleaning squid," she says.) Though the succession issue looms large at Michelin three-star royal houses--and usually involves only fathers and sons--Juan Mari was tactful enough not to insist on Elena's becoming a chef. "But at 17 I couldn't think of anything better to do with my life," she says a little apologetically. Rather than keeping her by his side, Juan Mari dispatched her abroad, first for a diploma from a fancy Swiss hostelry school, then for stints with members of the Michelin all-star fraternity (Alain Ducasse's Louis XV, in Monte Carlo; Troisgros, in Roanne, France).

"Here I was in 1989," Elena muses, "doing this classic French food at La Gavroche, in London, thinking how sophisticated it was--totally out of touch with what was happening at Arzak and in the rest of Spain." What was happening in Spain was Ferran Adrià, the Catalan visionary who forever changed the way Spanish chefs look at food. Anyone else might have felt threatened, but not Juan Mari (a man as shrewd as he is talented). Instantly he embraced Adrià and later sent Elena to do a stage at Adrià's restaurant El Bulli. "The entire time with Ferran, I thought I was hallucinating," Elena remembers. Yet, having inherited her father's gift for translating cutting-edge ideas into food with an intensely personal touch, she didn't return to San Sebastián an Adrià acolyte. What she wanted was to reconnect with her Basque roots. "I knew nothing about my culture's cuisine, so I tasted endlessly and read old cookbooks," she says.

"Like her father, Elena belongs to San Sebastián," says Oscar Caballero, a Paris-based food critic for the Spanish magazine Club de Gourmets. How lucky for her. A hotbed of innovative cuisine even in the world of avant-garde Spanish cooking, the San Sebastián area boasts 13 Michelin stars, 200,000 food-crazy inhabitants and three generations of astoundingly creative chefs. The founding fathers of Basque nouvelle cuisine--Juan Mari, Pedro Subijana of Akelare--are invigorated by the baby boomers, like the virtuoso Martín Berasategui (his eponymous restaurant just earned a third star), who in turn mentor the conceptually minded post-Adrià generation. "We are a family," Elena says. "We exchange recipes and ideas, help each other with banquets."

For all the critical praise--and recently the prestigious title of Chef de l'Avenir, given to Europe's most promising rising stars--Elena's adamant about acknowledging her debt to her father. "Paradoxically, his greatest influence on me was to leave me alone to develop my own vision," she explains. "I went through a phase of sticking bay leaves into everything, and he'd just smile." Today, the two are so close that "not even Elena and Juan Mari know where one ends and the other begins," says the critic Caballero. Juan Mari doesn't cook as much these days. But when father and daughter repair to their "lab" (a small kitchen above the restaurant, crammed with drawers full of edible exotica), she presents her ideas and dishes and then they discuss. And discuss. She's into agar and chamomile flowers; he goes for hake and parsley. "He always wants to clutter the plate. I like clarity," Elena says, laughing. He's as ebullient as she is composed (watching her at the stove, you'd never know that she's into black leather jackets, scooters and rock concerts).

Do they fight? "Like cat and dog," Juan Mari says gleefully. "But never for real." "He has the sharpest palate in the world," she gushes. "She opens new horizons for me," he returns.

Isn't it daunting for a woman to command a 35-person landmark kitchen, I ask Juan Mari as Elena goes on her rounds of the dining room. "Not if she's Basque," he retorts. "We are a matriarchy." "What about San Sebastián's famous no-women-allowed gastronomic societies?" I protest. He gives me one of his don't-you-just-adore-me smiles: "There's got to be some place a guy can escape from the domineering embrace of his wife."

Juan Mari lingers at my table, observing Elena as she introduces her latest creation to a table of regulars. "Look at their faces--they love it!" he whispers, loud enough for the entire restaurant to hear. I don't think I've ever seen a happier man.

Restaurante Arzak, 21 Alto de Miracruz, San Sebastián; 011-34-943-28-55-93.

Anya von Bremzen is the author, with John Welchman, of Please to the Table: The Russian Cookbook and of Terrific Pacific Cookbook.