Flatten each chicken breast half to 1/4-in. thickness. Place flour in a shallow bowl. Dip chicken in flour to coat both sides; shake off excess. , In a large skillet, brown chicken in 2 tablespoons butter on both sides. Transfer to a greased 11x7-in. baking dish. In the same skillet, saute mushrooms in the remaining butter until tender. Add the broth, salt and pepper. Bring to a boil; cook until liquid is reduced to 1/2 cup, about 5 minutes. Spoon over chicken., Bake, uncovered, at 375° until chicken is no longer pink, about 15 minutes. Sprinkle with cheeses and green onions. Bake until cheese is melted, about 5 minutes longer.