Combine first 6 ingredients. Divide mixture into 8 equal portions, shaping each into a 5-inch oval patty. Top each of 4 patties with 2 tablespoons cheese and 1/2 ounce ham, leaving a 1/2-inch border; top with remaining patties. Press edges together to seal.
Place patties on grill rack coated with cooking spray; grill 3 minutes on each side or until done. Place bread slices on grill rack; grill 1 minute on each side or until toasted. Top each of 4 bread slices with 1 lettuce leaf, 2 onion slices, 1 patty, 2 tomato slices, and 1 bread slice.