Whisk oil, 2 tablespoons vinegar, and 1 garlic clove in small bowl for marinade. Using spoon, scrape out gills from mushrooms and place mushrooms on rimmed baking sheet. Brush marinade over both sides of mushrooms, arrange hollow side up, and let stand at room temperature 30 minutes.
Prepare barbecue (medium heat). Mix panko, next 5 ingredients, and remaining 2 garlic cloves in medium bowl. Drizzle butter and remaining teaspoon vinegar over panko mixture and toss. Divide panko mixture among mushrooms, leaving 1/2-inch border around edges and packing down slightly. Place mushrooms on grill, stuffing side up; cover grill and cook until cheese melts and juices bubble at edges of mushrooms, rearranging mushrooms occasionally for even cooking (do not turn over), about 6 minutes.