Heat oven to 375°F. Lightly grease a 9x13-inch baking dish.
Boil noodles in salted water until just shy of al dente, 5–6 minutes. Drain and set aside.
Pat chicken dry. Season with about 1 teaspoon salt and the black pepper.
Warm olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Brown chicken in two batches, 5–6 minutes total. Transfer to a plate.
Melt remaining butter in the skillet. Cook mushrooms and onion with a pinch of salt until browned and tender, 6–8 minutes. Stir in garlic for 30 seconds.
Sprinkle flour over the vegetables. Cook, stirring, for 1 minute to form a paste.
Pour in Marsala wine and scrape up browned bits. Simmer 2 minutes to reduce slightly.
Stir in broth, cream, Dijon, and thyme. Simmer, stirring, until lightly thickened, 3–4 minutes. Adjust seasoning with remaining salt and more pepper.
Combine noodles, browned chicken, sauce, Parmesan, and half the mozzarella in the baking dish. Toss gently to coat.
Top with remaining mozzarella. Bake until bubbling and lightly golden, 15–18 minutes.
Rest 5 minutes. Finish with parsley and a pinch of red pepper flakes if using. Serve warm.