Soak wood chips in water at least 30 minutes; drain.
Prepare a charcoal fire on grill; scatter wood chips over hot coals.
Whisk together oil and garlic; brush on both sides of onion and mushrooms.
Grill onion and mushrooms, covered with grill lid, over medium-high heat (350° to 400°) 4 minutes on each side or until tender. Grill bun halves, cut sides down, 2 minutes or until lightly browned.
Stir together mayonnaise and next 5 ingredients. Layer 6 bottom bun halves evenly with lettuce, onion, mushroom, mayonnaise mixture, and 2 tomato slices; top with remaining bun halves.
Prep: 20 min., Soak: 30 min., Grill:10 min.