Flatten chicken breasts between sheets of waxed paper to 1/4-inch thickness. Season with salt and pepper. Cut each cutlet into quarters.
Heat 2 tablespoons butter in large heavy skillet, or 1 tablespoon in each of 2 medium-size skillets over medium-high heat. Cook chicken 5 minutes or until no longer pink in center, turning over halfway through cooking. Transfer chicken to platter.
Add mushrooms to skillet; cook 2 minutes. Add onion, shallot, bay leaf, garlic and thyme; cook, stirring, for 1 minute. Sprinkle in flour, stirring to blend. Stir in broth and wine. Simmer 2 minutes. Whisk in remaining butter to thicken slightly. Add chicken; heat through. Season with salt and pepper. Remove bay leaf. Serve chicken with sauce spooned on top.