LOS ANGELES—Tim Walker, as a struggling comedy writer, came to appreciate the qualities of a reliable staple: the grilled cheese.

"It was the go-to cheap, easy-to-make sandwich," he says. But he also found it "a gateway to gourmet food." He experimented with a variety of preparations, including banana bread slices filled with ricotta. "There was a galaxy of possibilities though mostly grilled cheese was relegated to the children's menu in restaurants."

Over...