HEAT oil in large nonstick skillet on medium-high heat. Add chicken; cook 6 to 8 min. on each side or until done (165ºF). Remove chicken from skillet, reserving drippings in skillet; cover chicken to keep warm.
ADD mushrooms and onions to drippings; cook 10 min., stirring occasionally. Add broth, garlic and thyme; stir. Cook 5 min.; stir in cheese.
TOP chicken with mushroom mixture and parsley.