Begin by making the Mornay sauce. Heat a small pot over medium-high heat. Once hot, add the butter. Once melted and bubbling, add the flour and whisk to combine. Cook, stirring constantly, for 2 minutes to cook the flour.

Gradually add the milk and continue to stir until the sauce thickens, about 2 to 3 minutes. Remove from the heat and add the gruyère cheese, dijon, nutmeg, and black pepper. Whick to combine and set aside.

Preheat the oven to 375 degrees F.

Lightly toast 6 slices of bread in a toaster. Butter 4 slices of the toasted bread, using 1 teaspoon per piece of toast. Set aside.

Lay two slices of bread butter side down on a baking tray. Spread 1 teaspoon of dijon mustard across each piece of toast. Top with a slice of ham, a heaping two tablespoons of the Mornay sauce, and sprinkling of gruyère cheese.

Place the two unbuttered piece of toast on top of the gruyère cheese and continue the layering process, finishing with the final piece of toast butter-side up.

Place in the oven and bake for 6 to 8 minutes to melt the cheese.

Remove from the oven and turn on the broiler to the high setting. Press each sandwich down using a spatula to press all the layers together. Top each sandwich with more Mornay sauce and the rest of the gruyère cheese. Place back in the oven and broil for 2 to 3 minutes, or until the cheese is golden and bubbling.

To cook the eggs, add the olive oil to a small frying pan over medium-high heat. Add the eggs and fry them sunny-side up. Season with salt and pepper.