Cook mushrooms in 2 tbsp butter in a large skillet, stirring constantly, just until tender.
Remove from heat; set aside. Cut each chicken breast half in half lengthwise. Place each piece of chicken between two sheets of wax paper; flatten to 1/8" thickness, using a meat mallet or rolling pin.
Dredge chicken pieces in flour.
Place 5 or 6 pieces of chicken in 1 to 2 tbsp butter in a large skillet; cook over medium heat 3 to 4 minutes on each side or until golden.
Place chicken in a lightly greased 13x9" baking dish, overlapping edges.
Repeat procedure with remaining chicken and butter.
Reserve pan drippings in skillet. Sprinkle reserved mushrooms over chicken.
Add wine and broth to skillet. Bring to a boil; reduce heat, and simmer, uncovered, 10 minutes, stirring occasionally.
Stir in salt and pepper.
Pour sauce over chicken.
Combine cheeses and green onions; sprinkle over chicken.
Bake uncovered at 450 for 12 to 14 minutes.
Broil 5 1/2" away from heat 1 to 2 minutes or until browned.