Turn on the broiler in your oven.

In a small pot melt the butter over medium heat and add the flour. Whisk constantly until the roux turns a light brown.

Whisk in the milk and season with salt and nutmeg to taste. A pinch of both should be enough. You're looking for a thick creamy sauce. If it's too thick add some more milk, if it's too runny, let it reduce a couple of moments.

Set the sauce aside.

Lightly toast the slices of brioche (be careful with it, because brioche tends to burn pretty quickly).

Spread two teaspoons bechamel sauce on a slice of brioche, add two slices of ham and repeat with a second slice.

Finish the sandwich with the last slice and sprinkle thick with the grated cheese.

Put the sandwich under the broiler for 5-6 minutes or until the cheese has melted.

Meanwhile fry the egg sunny side up.

Take out the sandwich and top with the fried egg. Serve immediately.