Cook mushrooms in 2 tbsp butter in a large nonstick skillet over medium-high heat, stirring constantly, 3 to 5 minutes or just until tender. Remove from heat; set aside.
Cut each chicken breast in half lengthwise. Place chicken between 2 sheets of heavy-duty plastic wrap; flatten to 1/8 inch thickness, using a meat mallet or rolling pin.
Dredge chicken pieces in flour. Cook chicken in batches, in 1 to 2 tablespoons butter in a large nonstick skillet over medium heat 3 to 4 minutes on each side or until golden. Place chicken in a lightly greased 13 x 9 inch baking dish, overlapping edges. Repeat procedure with remaining chicken and butter. Reserving pan drippings in skillet. Sprinkle mushrooms evenly over chicken.
Add 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 degrees for 12 to 14 minutes until cheese melts.