Wash and dry the chicken pieces. Beat the egg, 1 1/2 t salt and 1/4 t pepper. Dip chicken in mixture, then roll in the bread crumbs mixed with the cheese. Melt 4 tablespoons butter in a skillet, and sauté the chicken until browned and tender. Transfer to a baking pan and place in a 350 oven while preparing the sauce.
Melt the remaining butter in the skillet in which the chicken was cooked. Sauté the mushrooms 5 minutes. Blend in the flour, dash of salt and pepper. Gradually add the milk stirring steadily until it reaches the boiling point. Mix in the wine and cook over low heat 5 minutes. Arrange the chicken on a hot serving dish ad pour sauce over it.