Make the white sauce: Melt butter in a saucepan over low heat. Blend in flour, 1/2 teaspoon salt and 1/8 teaspoon pepper. Stir until smooth. Add milk; cook, stirring constantly, until mixture thickens and begins to bubble.
Wash asparagus and cut in 1/2-inch pieces. Cook asparagus in a small amount of boiling water until tender, about 5 minutes. Drain, reserving the cooking liquid. Set aside a few of the asparagus tips or pieces aside for garnish, if desired. Mash or blend remaining asparagus; set aside. Add enough boiling water to cooking liquid to make 1 cup; add white sauce and pureed asparagus. Heat thoroughly; season to taste with salt and pepper. Add whole asparagus pieces and serve.