Season chicken with salt and pepper, dredge lightly in flour, shaking off excess.
Heat 2 tbsp oil in a large skillet over medium-high, brown chicken 3–4 minutes per side until golden; remove and keep warm.
Add remaining oil, sauté mushrooms until they release liquid and brown, about 5 minutes.
Pour in Marsala, scrape browned bits, reduce by half, then add broth and simmer 3 minutes.
Return chicken to skillet, simmer 5 minutes until cooked through and sauce thickens. Stir in cold butter and parsley, adjust seasoning, then serve.