Melt the butter and the flour in a saucepan to make a roux. Stir on medium heat until smooth and bubbling. Do not let the roux brown.
Take the pan off the heat and add a little over half the milk. Return the pan to the heat and bring to the boil while stirring vigorously with a whisk. Add the remaining milk until the desired consistency has been reached.
Reduce the heat to low and simmer for ten minutes, stirring occasionally. Add the grated nutmeg. If the white sauce is to used as it is, add a little salt to taste.
To the basic Béchamel sauce, add freshly grated cheese. Fold in gently. Do not stir - the cheese may become stringy. Taste the sauce before adding salt - some cheese is salty, some is not.