Melt butter in a sauce pan over medium heat; saute mushrooms until browned. Add flour and stir until well blended. Continue stirring for an additional 3-4 minutes.
Using a whisk, slowly add warm milk and chicken broth, constantly whisk to avoid lumps. Sauce will begin to thicken. When sauce comes to a boil, turn heat down to very low and simmer for five minutes. Salt and pepper to taste. Makes 1 cup of sauce.