Place chicken breasts between two sheets of plastic wrap and pound to an even 1/2-inch thickness with a meat mallet. Cut into 4 equal portions. Mix flour with 1/2 teaspoon salt and 1/4 teaspoon pepper in a shallow dish. Dredge chicken in flour mixture, shaking off excess.

Heat 2 tablespoons olive oil and 1 tablespoon butter in a large oven-safe skillet over medium-high heat until shimmering. Add chicken in a single layer (work in batches if needed) and cook until golden brown, 3-4 minutes per side. Transfer to a plate; do not wipe the pan.

Add remaining 1 tablespoon oil to the skillet. Add mushrooms and cook undisturbed for 2 minutes to develop color, then stir and continue cooking until browned and their moisture has evaporated, about 6 minutes total. Season with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper.

Add remaining 1 tablespoon butter and garlic to the mushrooms; cook until fragrant, 30 seconds. Pour in Marsala, scraping up any browned bits from the pan bottom. Simmer until reduced by half, about 3 minutes. Add broth and thyme, then simmer 2 minutes more until slightly thickened.

Return chicken and any accumulated juices to the skillet, nestling into the sauce. Top each piece with mozzarella, then sprinkle Parmesan over all. Transfer to a 400F oven and bake until cheese is melted and bubbling, 10-12 minutes. Rest 3 minutes before serving.

Garnish with chopped parsley and serve directly from the skillet with crusty bread or buttered egg noodles to catch every drop of sauce.