Melt the butter over medium-low heat. Add the flour and whisk constantly for about 2 minutes, until thick and bubbly and just starting to lightly brown.

Slowly add the milk a little at a time while whisking constantly to prevent clumping. Increase the heat to medium and continue to whisk for 3-4 minutes until thickened and nearly boiling.

Add salt and pepper to taste as well as 1 pinch of nutmeg. Continue to whisk and cook for another 3-4 minutes.

Remove from the heat and set aside.

Set the oven to broil on high. Heat a large skillet over medium high heat.

Spread about 2-3 tablespoons of the butter evenly across each piece of bread, just one one side. Brown each piece of bread on the skillet on the buttered side, about 2-3 minutes.

Place two pieces of your browned bread, toasted side down into a baking dish. Spread the dijon mustard on the un toasted side of each piece.

Add about ¼ cup of the shredded gruyere cheese onto each of the slices of bread. Pour some of the bechamel sauce onto the grated cheese, about 3 tablespoons on each.

Add 5 slices of the ham onto the sauce covered cheese, you can add more to each if you want more meat.

Spread another 2 tablespoons of bechamel sauce onto the ham and top with the remaining 2 slices of bread, with the toasted side out.

Spread the remaining shredded gruyere cheese on both sandwiches and top with another 2 tablespoons of bechamel sauce.

Place the cheese covered sandwiches into the oven, on the center rack. Broil for about 2-3 minutes, or until the cheese has melted and has become browned and bubbly on top. Don’t over broil the sandwiches.

While the sandwiches are in the oven, heat the remaining butter in the skillet and fry the eggs, sunny side up. Add salt and pepper to taste.

When the sandwiches have come out of the oven, top them with the fried eggs, plate the sandwiches and serve with additional bechamel sauce.