Melt butter in large saucepan over medium heat. Whisk in flour, and cook and stir 3-5 minutes, or until it starts to turn golden. Add milk slowly, stirring as it thickens. Tear beef into bitesize pieces and add to sauce and season to taste with salt and pepper.
Spoon over toast, waffles, or split biscuits. If desired, top with slices or warm hard-cooked eggs.