For salsa, in a medium saucepan cook sweet peppers, onion, and ginger in hot oil over medium heat for 3 minutes. Add mango, jelly, lime juice, and salt; cook and stir until jelly melts. Set aside.
In a large bowl combine egg, 1/4 cup of the jerk sauce, and the bread crumbs. Add beef; mix well. Shape meat mixture into four 1/2-inch-thick patties.
For a charcoal grill, grill patties on the rack of an uncovered grill directly over medium coals for 10 to 13 minutes or until done (160 degrees F), turning and brushing once with the remaining jerk sauce halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place patties on grill rack over heat. Cover; grill as directed.)
Divide cheese among bottoms of rolls. Serve burgers on rolls with some of the salsa. Pass the remaining salsa.