French Breakfast Sandwich (Croque Madame)

4 tbsp butter
3 tbsp flour
2 cups whole milk
½ tsp SIDS SALT & PEPPER
⅛ tsp nutmeg
1 cup grated Gruyère cheese
¼ cup grated Parmesan cheese, plus more for garnish
8 thick-cut slices of bread
4 tsp Dijon mustard
4 eggs
16 slices of prosciutto
1 tbsp finely diced chives

Heat a saucepan over medium and melt 3 tablespoons of butter. When the butter has melted, add the flour and stir together. Cook, while continuing to stir, for 1 minute. Add milk and continue stirring until your roux has incorporated and the milk thickens, about 5 minutes. Remove from the heat.
Add SIDS SALT & PEPPER, nutmeg, cheese, and stir to combine. Stir the sauce every minute or so, until the cheese has melted then keep warm.
Preheat your griller.
Lay 4 pieces of bread on a baking tray. Spread 1 teaspoon of Dijon on each piece. Top with 2 pieces of prosciutto and 1 tablespoon of sauce. Cover with a second piece of bread and coat the top with 2-3 tablespoons more sauce.
Cook under a grill until the sauce is melted, golden and bubbling, 2-3 minutes. Turn off the grill and transfer sandwiches to the lower rack of the oven to keep warm.
While the sandwiches are cooking, heat a remaining tablespoon of butter in a pan over medium. Crack eggs and fry until the whites are set but the yolks are runny or to your taste.
Top the sandwiches with an egg and 2 pieces of prosciutto. Grate some Parmesan cheese and sprinkle on chives.

What Is The Difference Between A Croque Monsieur And A Croque Madame?
A croque madame is simply a croque monsieur with a poached or fried egg on top. The difference between the two sandwiches has to do with how the egg is incorporated into the rest of the ingredients: In a croque monsieur, the bread is dipped into the beaten egg before it's cooked (French toast-style).