1. Combine mayonnaise, cheese, lemon juice, garlic, pepper, and salt.
2. Brush eggplant slices (after slicing) on both sides with 1 tablespoon oil. Season with salt and pepper.
3. Grill, turning once, 8 minutes or until slightly charred and tender.
4. Brush tomatoes with remaining 1 tablespoons oil and season, if desired, with salt and black pepper. Grill, turning once, 4 minutes or until slightly softened. Remove and slice into quarters lengthwise.
5. Evenly spread bread with mayonnaise mixture, then top with basil, onion, eggplant and tomatoes.