Mr. Hong, 28, also has a record of revitalizing Asian classics. In 2014, he became the executive chef of an outpost of the barbecue chain Kang Ho Dong Baekjeong in Manhattan’s Koreatown, a neighborhood that he and his Korean-American friends considered culinarily moribund. “They made fun of me,” he said.

But the restaurant soon caught the attention of food-world luminaries like Anthony Bourdain and had to install a bouncer, with waits of up to two hours, before Mr. Hong decided to decamp for San Francisco.

For the museum, he said, he didn’t want to assemble “the Top 100 hits of Asian food.” His cooking is rooted in a dual heritage: The son of South Korean immigrants, he split his childhood among Texas, Alabama and New Jersey. He envisioned a menu that was nostalgic — “The best feedback to hear is ‘I grew up eating this,’” he said — but not hidebound by tradition, and his partners agreed.

“I think there’s a third culture,” Mr. Chen said, that arises when children are raised in a culture different from their parents’. “We see it as more of a remix, rather than, ‘Hey, this is authentic.’ It gives you a license to create.”

By August, the three men had made it through the first round of pitches to the museum. Now Mr. Hong was going to cook for Mr. Xu and his colleagues. The plan was to plate the food until the last course, when they would wheel in the cart. It couldn’t be just for show — they had to be able to lift the cart’s lids and reveal mantou (steamed buns) for dipping into condensed milk, matcha and caramel.