Heat the coals or grill. Spread teriyaki sauce over the mushrooms and marinate while the coals are getting ready.
Grill the mushrooms about 6 inches from the heat until tender (about 4-5 minutes a side).
Add the cheese on top and continue to grill briefly to melt cheese.
Assemble burgers by placing the lettuce leaf and tomato on the bottom bun then top with the cheese-topped mushroom. Spread the top bun lightly with half of the garlic mayonnaise and place on top of the mushroom (the lettuce keeps the bottom bun from getting soggy from some of the mushroom juice).