Begin by preparing your béchamel sauce. Heat Milk (1 1/2 cup) in a saucepan and in a separate saucepan, melt 1 tablespoon of the Unsalted Butter (1/4 cup).
Once milk is warm and butter is melted, add All-Purpose Flour (2 tablespoon) to butter and whisk for a minute or two.
Slowly add milk to butter-flour while whisking continuously until all milk is added and sauce is smooth and lump-free. Bring to a boil, but them bring to a simmer for a few minutes.
Season with Salt (to taste) and a pinch of Ground Nutmeg (to taste) and cook down for a few minutes, then set aside.
Spread a very thin layer of Dijon Mustard (1 tablespoon) on both slices of the Rustic Bread (2 slice).
Then add the Gruyère Cheese (2 slice) on each side and Ham (3 slice) in the middle.
Melt a pat of butter on your griddle or skillet over medium heat, and grill assembled sandwich until cheese is melted and both sides of the bread are toasty and delicious.
Spread béchamel on top of the sandwich and heap Gruyère Cheese (1/2 cup) on top all the way to the edge of the bread. Place in the broiler until cheese is melted and caramelized – but don't let it burn!
Meanwhile, heat ½ tablespoon butter in a small skillet or egg pan over medium heat. Gently crack Egg (1) into the pan, being careful not to break the yolk. Cook until white is just done and not snotty. Sprinkle with salt and place on top of the sandwich.
Voilà and bon appetit!