Pat chicken dry and season with salt, dredge in flour with salt and pepper 1 piece at a time, shaking off excess. In a large heavy skillet heat oil over moderately high heat until hot but not smoking and brown chicken in 2 batches, transferring with tongs to a large plate as browned.
Discard all but 1 tablespoon fat from skillet and sauté onion, mushrooms, stirring occasionally, until liquid mushrooms give off is evaporated. Add Marsala and cook mixture, stirring, bring to boil, reduce by 1/3. Add butter and chicken with any juices that have accumulated on plate and simmer, turning chicken once, until cooked through, about 5-6 minutes. Transfer chicken with tongs to a platter. Stir in thyme, and season with salt, pepper to taste. Transfer meat to a plate and pour the mushroom sauce over it, and serve.