In a food processor, pulse bread until fine crumbs form. Transfer to a medium bowl; add turkey, cheese, and onion. Season with salt and pepper, and mix gently just until combined. Form twelve 2-inch patties (about 3 tablespoons each).
In a large nonstick skillet, heat oil over medium heat. Cook patties until browned and cooked through, about 5 minutes per side. Serve on rolls with lettuce, tomatoes, ketchup, and mustard, if desired.