Lightly brush grill rack and hinged grill basket with vegetable oil. Then prepare grill with medium-hot coals, or heat gas grill.
Toss red pepper and onion with 2 tablespoons dressing in a bowl.
Coat chicken with 2 tablespoons dressing. Pound chicken between sheets of plastic wrap with flat side of meat pounder or bottom of skillet to 1/4-inch thickness. Refrigerate.
Grill pepper strips and onion in basket until tender and browned, 2 to 3 minutes per side. Coarsely chop. Toss with remaining dressing in bowl.
Remove the plastic wrap from chicken. Cook on grill until no longer pink in center, about 2 minutes per side.
Place 1/4 cup of cheese on each breast. Cover grill; cook until cheese melts. Place chicken in rolls, and top with vegetables.
