Slice chicken breasts into pieces about 3/4 inch thick. Combine flour, 1/2 teaspoon black pepper, 1/4 teaspoon cayenne pepper, and salt in a 1 quart plastic freezer bag. Mix milk and egg in a bowl. Dip chicken pieces in milk and egg mixture, and dredge in bag to cover with flour.
In fry pan, melt 1/2 stick of butter and olive oil. Sauté chicken over medium to medium high heat until a golden brown, being careful not to strip off flour coating when turning. Place chicken aside and keep warm.
Place onions and green pepper in fry pan and fry until onions are almost translucent. Add mushrooms and sauté until mushrooms have absorbed fat, then released it. Add garlic last. Then add wine and stir so as to deglaze fry pan. Add remaining black pepper and cayenne pepper to taste. Add remaining 1/2 stick of butter. Stir until the sauce thickens.
Return chicken to fry pan, and coat thoroughly with the sauce. Dish may be kept warm at this stage until ready to serve.
Serve hot on a large platter. Sprinkle with shredded parmesan cheese.
Preparation time: less than 1/2 hour.