Preheat your oven to 400F degrees.

In a small sauce pan, over medium heat, melt the butter and then add the flour, stirring constantly for about 2 minutes.

Slowly add the warm milk, whisking until the sauce has thickened.

Turn the heat off and season the sauce with salt, pepper and nutmeg.

Add 1/2 cup Gruyere and 1/2 cup Parmigianino Reggiano and stir until the cheese melts and incorporates into the sauce. Reserve.

Place English muffins with the inside sides up on a baking sheet covered with parchment paper.

Add 1/2 tablespoon of mustard to one side of each muffin. Then add 1 to 2 tablespoons of the Mornay sauce to the other 6 sides.

Top the mustard sides with the ham (cut the slices of ham in small pieces so they can fit in the muffin) and 1/4 cup of grated Gruyere cheese each.

Top with the remaining muffin slices.

Spoon about 1 tablespoon of Mornay sauce on top of the sandwiches and finish with the rest of the Gruyere cheese.

Bake the sandwiches for 5 minutes.

Turn on the broiler and broil the sandwiches until the top is bubbly and slightly browned.

While the sandwiches are in the oven, you can start cooking the eggs.

Heat a little olive oil in a large skillet, over medium heat.

Add the eggs and cook until the whites are firm but the yolks are still runny. Season with salt and pepper.

Top the sandwiches with the eggs and serve hot!