Melt 2 tablespoons butter in skillet and then add sliced mushrooms, cooking until just tender. Remove from pan and set aside.
Place each chicken breast between plastic wrap and flatten to 1/4 inch using a meat mallet.
Dredge each breast in flour to lightly coat.
Melt 2 tablespoons butter in same skillet and add half of breasts. Saute until golden about 4 minutes each side.
Repeat with remaining 2 tablespoons of butter and the other 3 breasts.
Place breasts in a lightly greased 9x13 pan. Top with the mushrooms.
To the skillet add the wine and chicken broth. Stir up drippings from pan and bring to a boil. Add the salt and pepper. Decrease heat and simmer, uncovered 8 minutes, stirring occasionally.
Pour sauce over chicken and mushrooms.
Mix the cheeses together and sprinkle on the chicken.