Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add mushrooms and onions; cook 8 minutes or until lightly browned and liquid is absorbed, stirring frequently. Sprinkle with 1/4 teaspoon each of salt and pepper. Transfer mushroom mixture to a plate.
Sprinkle chicken pieces with flour, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Toss lightly, and coat with cooking spray. Add remaining 2 teaspoons oil to pan; add chicken, and cook 3 minutes on each side or until browned on each side. Push chicken to one side of pan, and add wine; cook 2 minutes or until liquid evaporates, scraping pan to loosen browned bits. Return mushroom mixture to pan; add broth. Cover, reduce heat, and simmer 5 minutes or until chicken is done. Stir in thyme.