Pre-heat broiler with rack in the top position.

Melt 2 tablespoons of the butter in a medium saucepan over medium heat. Whisk in flour and cook, whisking until the mixture bubbles, about 1 minute. Pour in milk while whisking constantly to ensure there are no lumps. Bring mixture to a simmer and cook, stirring often, until the mixture thickens, 8 to 10 minutes. Remove from heat and add the parmesan and half of the gruyere cheese; stir until cheese has melted and the mixture is smooth and creamy. Add the nutmeg and taste and adjust seasoning with salt and pepper. Set aside.

On a clean work surface arrange 4 slices of the bread and spread each with a teaspoon of the Dijon. Evenly lay 2 slices of ham on top of each slice and top with generous tablespoon of sauce, spreading it slightly out towards the edge. Evenly sprinkle the remaining gruyere cheese over the sauce and top each of the remaining bread slices.

Heat a large heavy skillet over medium heat and add 2 tablespoon butter to the skillet; melt until butter is foaming. When the foaming subsides place two sandwiches in the skillet and cook until golden on one side and cheese is beginning to melt, about 2 minutes, turn the sandwiches over and cook on the other side, about 1 minute more. Transfer to a foil-lined rimmed baking sheet and spread generously with sauce. Repeat with remaining sandwiches, wiping out skillet and adding another 2 tablespoons butter. Transfer baking sheet to the oven and broil until the sauce is golden-brown and bubbling, 3 to 4 minutes. Remove from the oven and keep warm while preparing the eggs.

Heat a large non-stick skillet over medium heat and add remaining 2 tablespoon butter to the skillet; melt until butter is foaming. When the foaming subsides, add the eggs and cook until just set, about 3 minutes. Place one egg on top of each sandwich, season with salt and pepper and serve immediately.

Photo by: Karlee Flores