Melt the butter in a small saucepan over medium heat. Stir in the flour with a wooden spoon and continue stirring for 1 minute.
Using a whisk, gradually stir in the milk. Cook over medium heat, stirring constantly to prevent any lumps, until the sauce is smooth and thickened to the texture of heavy cream, about 3 minutes. Reduce the heat to low and continue simmering until the flour loses its raw flavor, 2 minutes. Season with the salt, pepper, and nutmeg.
Use at once or cover with a film of milk or melted butter, cover with plastic wrap, and refrigerate.