Place butter in a clean saucepan over medium heat and allow to melt, but do not allow to brown.
When the butter bubbles, add the flour and using a wire whisk blend it into the butter.
Whisk flour and butter together for a between 1 minute and 2 minutes, giving the flour a chance to cook. It will form a glutinous ball.
Add the milk gradually, a little at a time, whisking all the time to make sure it incorporates evenly and there are no lumps.
Once all the milk is used, add ¼ cup of the grated Parmesan, or ½ cup of the strong cheddar cheese, a generous pinch of salt and the nutmeg. Keep whisking to make sure no lumps form at this stage.
Keep whisking (yes, there is a lot of whisking involved), as you cook the sauce for around 3 minutes.
Turn off heat, remove skillet from the hot plate, and allow to cool slightly before folding in the two beaten egg yolks (optional).