Begin by preparing your béchamel sauce. Heat milk in a saucepan and in a separate saucepan, melt 2 tablespoons of the unsalted butter. Once milk is warm and butter is melted add flour 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 then bring to a simmer for a few minutes. Season with salt, pepper and a pinch of nutmeg and cook down for a few minutes, and then set aside.
Spread a very thin layer of mustard on both slices of bread then add sliced Gruyere on each side and ham in the middle. Melt 1 teaspoon of butter on a skillet pan over medium heat, and grill assembled sandwich until the bread toasty. Flip the sandwich, add 1 teaspoon butter and grill the other side. Spread béchamel on top of the sandwich and heap shredded Gruyere on top all the way to the edge of the bread. Place in the broiler until cheese is melted and caramelized. Be careful don’t let it burn!
Meanwhile, heat 50 gr./ 1.76 oz. olive oil in a small skillet or egg pan over medium heat. Gently crack egg 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.
