Light 1 side charcoal grill or preheat gas grill to 350° to 400° (medium-high); leave other side unlit. Toss together peppers and oil. Arrange peppers in grill basket or on aluminum foil tray over unlit side; grill, covered with grill lid, 10 to 15 minutes or until peppers begin to shrivel. Transfer peppers to lit side of grill, and grill, covered with grill lid, 8 to 10 minutes or until lightly charred, turning after 4 minutes. Remove from grill to wire rack, and cool completely (about 15 minutes). Remove and discard stems; slice peppers in half lengthwise. Remove seeds, and sprinkle peppers with salt to taste.
Light other side of grill, and preheat to 350° to 400° (medium- high). Shape ground chuck into 12 patties; sprinkle with desired amount salt and pepper. Grill patties, without grill lid, 4 to 5 minutes on each side or to desired degree of doneness. Place 2 to 3 pepper halves on each patty; top with cheese. Grill, covered with grill lid, 1 to 2 minutes or until cheese melts.
Butter inside of each bun, and toast on grill. Serve burgers on toasted buns with desired toppings.