1. Sprinkle eggplant lightly with salt; place in colander and let stand 30 minutes. Rinse well; drain and pat dry. Meanwhile, heat grill to medium-high direct heat. Assemble eggplant, onion. bell pepper, and garlic clove on large baking sheet. With 3 tbsp. olive, brush 1 side of each with olive oil.
2. Place vegestables, oil side down, on grill. Brush tops lightly with oil. Grill covered 6 to 8 minutes or until vegestables are grilled and tender but not limp, turning once. Remove vegestables; grill rolls cut side down for 1 minute.
3. Peel garlic; mash or chop. Add to mayonnaise, along with remaining 2 tsp. olive oil and basil in small bowl. Spread cut sides of each roll lightly with mayonnaise mixture. Top each sandwich equally with eggplant, onion, tomato slices, bell pepper, and 1 slice cheese.
4. To melt cheese, wrap each sandwich loosely in heavy duty foil. Return to grill and cover; heat 5 to 8 minutes, turning once.