Saute mushrooms in 2 T. butter and set aside.
Slice each chicken breast into two thinner filets and place between 2 sheets of plastic wrap. Using a mallet, flatten to 1/4 inch thickness..
Dredge each chicken filet in flour and saute each side until golden brown, about 3-4 minutes.
Add the cooked chicken to a 9 x ⅔13-inch baking dish, overlapping edges. Before adding the chicken, line the baking dish with aluminum foil. Reserve the pan drippings in the skillet.
Sprinkle mushrooms over chicken. Then, add the cooking wine and stock to the pan drippings. Bring to a boil and simmer, uncovered for eight minutes. Stir in the salt and pepper.
Pour sauce evenly over the chicken. Combine cheese and green onions, and sprinkle over the chicken. Bake, uncovered, at 375 for 20 minutes.
Serve with sauce drizzled over the top.