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.