Preheat oven to 450F. Lightly oil a rimmed baking sheet.
In a small bowl, whisk together 2 tablespoons of the olive oil, the vinegar, 1 teaspoon of minced garlic, and salt and pepper. Brush over both sides of the mushrooms and set them cap-side-down on the baking sheet.
In a medium bowl, combine the remaining garlic, the panko bread crumbs, cheeses, and oregano and thyme. Stir in the last 2 tablespoons of olive oil. Divide the cheese mixture among the mushroom caps, scooping some into each and pressing it lightly.
Bake until the mushrooms are tender and the cheese is melted and golden brown, 15-20 minutes. Serve hot, and spoon some pasta sauce on at the table.