1. For sauce, combine all ingredients in saucepan. Cook, stirring over moderate heat until sauce boils; simmer 10 minutes.
2. Preheat grill.
3. 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.
4. 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.
5. Place burgers on rolls, and spoon on remaining Pineapple-Papaya Sauce. Top with pineapple slice (if desired), lettuce, tomato, and onion slices (if desired).