A simple, four-ingredient sauce elevates this burger from pretty good to perfect. After eating this burger, you'll dream of luaus.
For sauce, combine all ingredients in saucepan. Cook, stirring over moderate heat until sauce boils; simmer 10 minutes.
For burgers, mix ground turkey, breadcrumbs, onion, cayenne, sugar, and 1/3 cup sauce in a bowl until just combined. Form into 4 (1-inch) patties.
Brush burgers with oil; season with salt and pepper. Grill over medium heat, turning once, about 6 minutes per side. Baste with sauce, and grill 2 minutes more; transfer to plate. If desired, grill pineapple slices 2 minutes per side; transfer to plate.
Place burgers on rolls, and spoon on remaining Pineapple-Papaya Sauce. Top with pineapple slice (if desired), lettuce, tomato, and onion slices (if desired).