The first time I tasted New Orleans-style barbecue shrimp, I was confused. It had none of the smoky char associated with food off a grill or smoker. And that’s because it’s not actually barbecue.

Every major city seems to claim a sandwich as its own: Chicago with Italian beef sandwiches, New York with pastrami on rye, Buffalo with beef on weck, Philadelphia with cheesesteaks. New Orleans, a ...