Cuban Bread, Ham, Swiss Cheese, Dijon Mustard, Garlic, Pickle Relish, Mojo Marinade - PRESSED. . . . . . . HISTORY: A Cuban Sandwich is a variation of a ham and cheese sandwich that likely originated in cafes catering to Cuban workers in Tampa or Key West, two early Cuban immigrant communities in Florida centered on the cigar industry. Later on, Cuban exiles and expatriates brought it to Miami, where it is also very popular.