Prepare a charcoal fire on grill.
Stir mayonnaise, basil, mustard, lemon juice, salt and pepper together; set aside.
Remove stem from mushrooms; clean. Place wooden skewers through onion slices to hold rings in place. Whisk together oil and garlic; brush on both sides of mushrooms and onions.
Grill onions and mushrooms, covered with grill lid, over medium-high heat four minutes on each side or until tender. Grill bun halves, cut side down 2 minutes or until lightly browned.
Layer 6 bottom bun halves evenly with lettuce, onion, mushroom, mayonnaise mixture, and 2 tomato slices; top with remaining bun halves. Yield: 6 servings.