Wash the asparagus thoroughly. Cut the tips into 1 1/2 inch pieces. Trim the rest of the asparagus stalks of any tough or woody parts and set aside. Place tips in saucepan and cover with water. Cook until tender, about 5 minutes. Remove tips and set aside. Cook remainder of asparagus in the same water until tender. Drain.
Allow cooked asparagus stalks to cool slightly and grate. Melt butter in a large saucepan over medium high heat. Stir in flour and cook, while stirring, a minute or two. Slowly, add milk, while stirring. Add grated asparagus and cook for 5 minutes. Season with salt and pepper.