Place mushroom caps, smooth side up, in a shallow dish.
Whisk together balsamic, oil, garlic, basil, and oregano in a small bowl. Season with salt and pepper.
Pour vinaigrette over mushrooms. Let stand at room temperature for 15 minutes or so, turning twice.
Preheat grill for medium-high heat.
Brush grill grates with oil. Place mushrooms on grill, reserving marinade for basting. Grill until just tender, 5 to 8 minutes per side, brushing with marinade frequently.
Top mushrooms with cheese and continue grilling until cheese is melted, about 2 minutes.
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