Clean mushrooms with a damp cloth and remove their stems. Place in a glass dish, stem (gill) side up.
To prepare the marinade, in a small bowl whisk together the vinegar, water, sugar, garlic, cayenne pepper and olive oil. Pour over the mushrooms. Cover and marinate in the refrigerator for about 1 hour, turning mushrooms once.
Prepare a hot fire in a charcoal grill or heat a gas grill or broiler. Away from the heat source, lightly coat the grill rack or broiler pan with cooking spray. Position the cooking rack 4 to 6 inches from the heat source.
Grill or broil the mushrooms on medium heat, turning often, until tender, about 5 minutes on each side. Baste with marinade to keep from drying out. Using tongs, transfer the mushrooms to a plate.
Place each mushroom on a bun and top with 1 tomato slice, 1 onion slice and 1/2 lettuce leaf. Serve immediately.