To make Parmesan shavings, start with a chunk of cheese thats at least 4 ounces. Use a swivel-edged vegetable peeler to shave off attractive curls, letting them fall onto a piece of parchment or wax paper.
Prepare a hot charcoal fire or preheat gas grill on high. Set a vegetable grill topper on rack to heat.
Vinaigrette: In small bowl, whisk together all vinaigrette ingredients along with 1 tablespoon water.
Place greens in large bowl and set aside. Spread mushrooms, onions, bell pepper and tomatoes on 2 baking sheets. Brush vegetables all over with oil and season with salt and pepper.
Lightly oil grill topper. Grill vegetables in batches until nicely browned and tender, 2 to 4 minutes per side. Transfer cooked vegetables to baking sheet. Grill bread alongside vegetables until lightly toasted. Immediately rub warm grilled bread with cut sides of garlic.
Add grilled vegetables and vinaigrette to greens and toss well. Divide salad among plates and sprinkle Parmesan shavings on top. Serve right away with garlic bread.