1. Heat broiler. Line broiler pan with foil. Place bell pepper on pan and broil 5 inches from heat 15 minutes, turning once halfway through, until evenly charred. Wrap pepper in foil and let stand until cool 15 minutes. Peel, seed, and cut pepper into 4 pieces; set aside.
2. Meanwhile, combine chicken, bread crumbs, cilantro, shallots, jalapeno, lime juice, and salt in a large bowl. Shape into four 1/2-inch-thick patties; transfer to a plate. Cover patties and refrigerate until firm, about 10 minutes.
3. Broil patties 15 minutes, turning once halfway through, until meat thermometer inserted in center of each patty reaches 160 degrees F. Top each patty with piece of roasted pepper; divide cheese and sprinkle over tops.
4. Broil patties 1 minute more, until cheese is bubbly. Place burgers on bottom halves of rolls; top with remaining halves. Garnish with salsa, tortilla chips, and lime wedges, if desired. Makes 4 servings.