You might reach for it at Chinese restaurants or consider it a household staple, right next to the mustard in your refrigerator: Asian chili crisps have exploded onto the mainstream market in the U.S.

Anastacia Quiñones-Pittman, the executive chef at José, pours one of her spicy salsas on a taco at the Dallas restaurant. Shafkat Anowar / Staff Photographer Jody Denton, a self-described “chili head, ...