1Preheat 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 (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.
2Broil (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 and serve immediately.