Remove stems from mushrooms; chop and reserve. Place mushroom caps on foil-lined baking sheet; drizzle with 1 Tbsp. oil.
Melt butter in large skillet. Add mushroom stems, onion and garlic and cook over medium heat 5 minutes, stirring occasionally. Remove from heat. Stir in breadcrumbs, 1 cup cheese and salt and pepper.
Fill mushroom caps with stuffing mixture; top with remaining cheese. Bake in preheated 400°F oven for about 5 minutes until cheese is melted. Serve warm.