Brush 1 tablespoon olive oil evenly over both sides of eggplant, onion, zucchini, and coat asparagus spears. Sprinkle with rosemary, pepper, and salt.
Combine 1 tablespoon oil and vinegar in a bowl. Brush vinegar mixture over cut sides of bread.
Place onion on grill rack coated with cooking spray, and grill for 6 minutes on each side or until tender. Remove from grill, and separate into rings.
Grill eggplant, zucchini, and asparagus 4 minutes on each side or until tender. Cut zucchini pieces in half crosswise.
Place bread, cut sides down, on grill rack; grill 2 minutes. Remove from grill.
Place 1 piece of cheese on bottom halves bread portions; top each serving with 1 eggplant slice, 1 tomato slice, 1 basil leaf, 2 pieces zucchini, one-fourth of onion rings, 2 asparagus spears, 1 eggplant slice, 1 tomato slice, 1 basil leaf, 1 piece of cheese, and top half of bread. Place sandwiches on grill rack; grill 2 minutes, covered, or until cheese melts.