Place the mushroom caps, smooth side up, in a shallow dish. In a small bowl, whisk together vinegar, oil, basil, oregano, garlic, salt, and pepper. Pour over the mushrooms. Let stand at room temperature for 15 minutes or so, turning twice.
Preheat grill for medium-high heat.
Brush grate with oil. Place mushrooms on the grill, reserving marinade for basting. Grill for 5 to 8 minutes on each side, or until tender. Brush with marinade frequently. Top with cheese during the last 2 minutes of grilling.
The nutrition data for this recipe includes information for the full amount of the marinade ingredients. Depending on marinating time, ingredients, cooking method, etc., the actual amount of the marinade consumed will vary.
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