Preheat 350 degree oven. In large saucepan over low heat, melt butter. Blend in flour and seasonings. Cook, stirring constantly until smooth and bubbly. Remove from heat. Stir in broth and evaporated milk. Heat to boil, stirring constantly. Boil and stir for one minute. Stir in wine, cooked chicken/turkey, drained mushrooms.
Pour all into a 2 quart casserole. Sprinkle top with cheese. Bake uncovered, for 30 minutes or until bubbly. Can brown by placing briefly under broiler.