If desired, pound chicken between two pieces of wax paper or plastic wrap (I like to use a rolling pin to do this); season with salt and pepper.
Heat the oil in a large skillet over medium-high heat; add chicken and cook until done, about 5 minutes per side. Remove from skillet and keep warm.
In the same skillet, add onions and mushrooms; cook until soft.
Add wine to pan and cook for 1 to 2 minutes. Stir in chicken broth. You may add more broth if it is too thick for your taste.
Pour sauce over chicken, sprinkle with chopped parsley and serve.