It had been a culinary nobody, mushy and maligned.

But when a chef as decorated as Daniel Humm turns his attention to perfecting a veggie burger, the signal is clear: That second-fiddle vegetarian staple has arrived.

Mr. Humm, whose restaurant Eleven Madison Park in Manhattan has three Michelin stars, serves a carefully constructed veggie burger at the NoMad Bar nearby. His version uses grains, fresh vegetables, quinoa and lentils “to replace the texture of meat,” he said, with eggs, cream cheese and Dijon mustard to hold it together. “Nothing goes into the burger that doesn’t serve a purpose,” Mr. Humm said.

“We wanted this to be able to stand next to our beef burger and our chicken burger, not be a dish we just put on the menu for the sake of it,” he added.

After decades as an amateur player eager for a big break, the veggie burger has made its ascent, becoming a destination dish and hashtag darling as never before.