Melt 1 tbsp butter in a saucepan over medium heat. Stir in 2 tbsp flour and whisk until smooth (about 1 minute).
Slowly pour in 1 cup milk, whisking constantly to avoid lumps. Cook until the sauce thickens (about 3 minutes).
Season with a pinch of nutmeg, salt, and pepper. Set aside.
Spread Dijon mustard on two slices of bread.
Layer each with 2 slices of ham and half the grated cheese.
Top with another slice of bread.
Spread softened butter on the outside of each sandwich.
Heat a skillet over medium heat and cook the sandwiches for 3-4 minutes on each side until golden and crispy.
Preheat your oven’s broiler.
Transfer sandwiches to a baking tray, spoon béchamel sauce over the top, and sprinkle with remaining cheese.
Broil for 2-3 minutes until the cheese is bubbly and golden.
Heat 1 tbsp vegetable oil in a skillet over medium heat. Crack 2 eggs into the pan and cook until the whites are set but yolks are runny (or as preferred).
Top each sandwich with a fried egg.
Serve hot with a simple side salad or fries for the full bistro experience.