In a shallow dish, whisk together vinegar, oil, basil, oregano, garlic, salt and pepper. Place the mushroom caps in the dish and turn to coat with the vinegar mixture. Let stand at room temperature for 15 minutes or so, turning twice.
Preheat barbecue for medium-high heat.
Place mushrooms on the barbecue, reserving marinade for basting. Cook for 5 to 8 minutes on each side, or until tender. Brush with marinade frequently. Top with optional cheese during the last 2 minutes of barbecuing.