Brush the mushrooms with olive oil, then grill or roast. Allow to cool.
Combine the cheeses until mixed well, then add rosemary and oregano. In a separate bowl, combine panko crumbs, garlic, and olive oil. Set aside.
Meanwhile, place the vinegar in a small saucepan and simmer until reduced by 3/4. Allow to cool.
Fill each mushroom with cheese mixture, top with bread crumb mixture, and bake at 375 degrees for about 10 minutes. Allow to cool slightly before handling.
To serve, plate mushrooms, sprinkle with salt and pepper, and drizzle balsamic reduction on top.
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