For the bechamel sauce: Place a small sauce pot over medium-low heat and melt 2 tablespoons of butter. Stir in the flour and cook for 1 minute or so. Whisk in the milk and bring it to a bubble then drop the heat to low. Season the sauce with salt, pepper, nutmeg and Dijon. When the sauce thickens enough to coat the back of a spoon, turn off the heat.
Heat 1 tablespoon of butter in a medium non-stick skillet over medium-low heat. When the butter melts, crack the eggs and add them to the pan, keeping the whites separate from each other. Cook to desired doneness, sunny-side-up to over-hard.