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 at least 15 minutes preferably longer, turning occasionally.
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.
Recipe by Living Full on Life at https://www.livingfullonlife.com/?p=1324