Add onion, thyme and the remaining 1/4 teaspoon pepper to the pan; cook, stirring often, until the onion is translucent, 2 to 3 minutes. Return the mushrooms to the pan. Stir in Marsala and broth, scraping up any browned bits. Simmer over medium heat until slightly reduced, about 2 minutes. Remove from heat and stir in butter. Serve the chicken topped with the sauce.