Place chicken between 2 sheets of waxed paper and pound until about 1/4 to 1/2 inch thick. Sprinkle with salt, pepper and flour. Quickly saute in 1/4 cup butter until golden. Remove to plate.
Add remaining 1/4 cup butter to pan and saute onion, garlic, and mushrooms slowly until cooked but not browned. Stir in flour, salt, pepper, chicken stock, and wine. Cook until thickened about 5 minutes. Stir in mashed avocado and 1/2 cup cheese.
Arrange chicken breasts in glass baking dish. Top with sauce, remaining cheese, and green chilies, if desired. Bake at 350°F for 10 minutes.