Preheat oven 400°.
Toss mushroom caps with 1 tablespoon oil in large bowl. Season, if desired, with salt and pepper. Arrange mushroom caps in 13 x 9-inch baking dish, stem-side-down. Bake 10 minutes or until golden brown. Remove mushrooms; drain liquid. Return mushrooms to baking dish, stem-side-up.
Heat remaining 1 tablespoon olive oil in 12-inch skillet over medium-high heat and cook shallots, stirring occasionally, 1 minute. Stir in mushroom stems and cook, stirring occasionally, 8 minutes or until tender. Stir in hot pepper flakes, parsley and thyme. Season, if desired, with salt and pepper. Turn into medium bowl. Stir in 1/4 cup Mayonnaise, 1 tablespoon vinegar, 2 tablespoons cheese and bread crumbs. Evenly spoon mixture into mushrooms.
Combine remaining 1/4 cup Mayonnaise with remaining 1 tablespoon vinegar in small bowl. Season, if desired, with salt and pepper. Drizzle over mushrooms, then sprinkle with remaining 2 tablespoons cheese. Bake 15 minutes or until mushrooms are golden and heated through.