The domestic terrorism that had long marred Spain’s Basque Country abated. San Sebastián — with its pintxo bars and spectacular seaside — became a magnet for foodies and foreign chefs. Mr. Rivera said the mix of tourism, marketing and universally available ingredients — supermarket cream cheese, eggs, sugar, some flour — led the cheesecake’s reputation to spread across borders and cultures.

As early as 2008, New York restaurant menus included a homage to the “Burnt Basque Cheesecake” — a name Mr. Rivera never embraced. “It’s not burned,” he said. Yet the cheesecakes hardly made a dent in a land where Junior’s, the local Brooklyn cheesecake heavy, loomed large.

Around 2012, though, popular bakeries in Turkey started featuring the San Sebastián cheesecake, clearly influenced by La Viña’s creation. In the following years, it began to colonize London and Chicago, Malaysia and Australia. By the time Bon Appétit, the American food magazine, published a recipe for it in 2019, Basque cheesecake was everywhere.

During the coronavirus pandemic, new legions of home cooks got creative with Mr. Rivera’s recipe, which he never kept secret, long demonstrating it on a DVD that he still hands out in his restaurant. “Today, we are approaching Peak Cheesecake,” the restaurant reservation service and guide Resy wrote in 2021. That was the year The New York Times predicted the Basque Burnt cheesecake would be the “flavor of the year.” In the meantime, Instagram ate the cake up. The Times of London in 2023 called it “the pudding that broke the internet.”

“I thought that it would stop being trendy,” said Amaia Ormazábal, 34, who came by La Viña to pick up a couple of whole 55-euro (or roughly $64) cheesecakes for a lunch with friends. She showed pictures of her wedding, featuring 20 “authentic” cheesecakes from La Viña and said she was opening her own restaurant in Madrid, where, she said, the demand for Basque cheesecakes remained sky high. “I’m going to have it on the menu,” she said.