When cleaning mushrooms, don't immerse them in water because they'll soak up the water like a sponge. Instead, wipe the mushrooms clean with a damp cloth or sturdy paper towel.
Heat the broiler (grill). Position the rack 4 inches from the heat source. Lightly coat a glass baking dish with cooking spray. Place the mushrooms in the dish, stemless-side (gill-side) up.
In a small bowl, whisk together the vinegar, brown sugar, rosemary and garlic. Pour the mixture over the mushrooms. Set aside for 5 to 10 minutes to marinate.
Broil (grill) the mushrooms, turning once, until they're tender, about 4 minutes on each side. Sprinkle grated cheese over each mushroom and continue to broil (grill) until the cheese melts. Transfer to individual plates.
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