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