Start by preparing the Mornay Sauce. In a medium sauce pan, over medium heat, melt the butter. Mix in the flour with a wooden spoon. Cook for 2-3 minutes, mixing continuously until the roux paste comes together. Slowly add in the milk and mix (using a whisk this time) until reaching a creamy consistency. Add the nutmeg, cheese, salt and pepper to taste, and whisk until all the cheese is melted and everything is combined into a creamy consistency. Remove from the heat and set aisde.

Arrange 4 slices of bread (of the same size) on your working surface. Spread ½ tbsp of butter on 2 slices. Top each of them with 2 slices of ham, and ¼ cup of grated cheese. Close each with the 2 remaining bread slices.

Heat a large cast iron skillet and melt the remaining butter into it (1 tbsp). Add the two sandwiches and brown on one side for 2-3 minutes. Turn on the other side (carefully, with a spatula) and brown for an additional 2-3 minutes. Pour Mornay sauce on top of the 2 sandwiches, and pass under the broiler for 1-2 minutes until the Mornay is lightly browned.

In the meantime, crack 2 eggs in a greased skillet and fry them up (sunny side up, or over-easy). Take the sandwiches out of the oven, top each carefully with a fried egg, and season to taste with salt, pepper and chopped chive.

Serve immediately.