Croque Monsieur à la Émilien

I ask your pardon for the post I missed, but the recipe that I prepared for Wednesday was fine but not fully convincing, it needs further testing and some improvements.
Today however I submit this one without fear because I'm devouring it as I'm writing now. While I traveled through France I ate a couple of these typical sandwich but I had never thought to prepare them once at home. Then some days ago I find one of the many recipes to prepare them and I realized that, even with a thousand variations, its main ingredients are grated Gruyère cheese, ham and béchamel. "Corbezzoli!" I thought "but it seems a recipe from Emilia. If only they'd use Parmesan cheese and prosciutto it may be called Croque Sgnàur (Monsieur, alas Sir, in dialect of Bologna ehehe)". So why not try to use these two ingredients in place of the originals?

Ingredients for 2 sandwiches: 4 slices of white bread without crust, 2 slices of prosciutto, ½ cup of grated Parmesan cheese, 1 egg, 2 and ½ oz. of butter, 1/3 cup of flour, ½ cup of milk, salt, pepper, nutmeg.

Preparation: sprinkle two slices of bread with half of the Parmesan, place on top of each one a slice of prosciutto, trying to keep it within the edges, and sprinkle over the remaining cheese. Close the sandwich with other two slices of bread and squeeze it well. Beat the egg and soak inside the sandwich. Melt 1 and ½ oz. of butter in a large frying pan and when it is hot, brown the sandwiches inside. Then prepare the béchamel sauce: melt over low heat the remaining ounce of butter, pour in gradually the flour using a sieve, so not to form lumps, stirring constantly. When the mixture begins to brown add milk, previously warmed, little by little, stirring constantly. When the sauce starts to get creamy, add two pinches of salt and grate in pepper and nutmeg according to your taste. Spread the sauce over each of the two sandwich and put them in the oven to grill for 3-4 minutes until the béchamel browns completely. Serve them hot.