In a medium saucepan over medium heat, melt butter and add flour. Whisk constantly for about a minute. After about 2 minutes or when the mixture becomes bubbly, slowly add in milk.

Keep whisking until thickened enough to coat the back of a spoon and can remain separated when you run a finger down it.

Season with salt, pepper, and whisk in dijon mustard.

Take off the heat, then add ½ cup of gruyere cheese and ¼ cup grated parmesan, whisking until the cheese is fully melted and the sauce is thick and creamy. If the sauce gets a little too thick, add a splash or two of milk and whisk to thin to the correct consistency. Set aside.

Preheat broiler on low.

Heat a cast iron skillet over medium heat spread butter onto one side of each slice of the bread, then layer on the ham and shredded cheese and sandwich between the bread slices. Brown the sandwich on each side for 2-3 minutes until golden brown.

Generously cover the sandwiches with more mornay sauce and all the rest of the gruyere.

Place the baking sheet with sandwiches under the broiler. Leave to cook for about 3 minutes or until the bechamel sauce is browned. Keep an eye on it, DO NOT LET IT BURN.

While the sandwiches are broiling, melt butter in a nonstick skillet. Crack two eggs into two separate small bowls and add them into the pan one by one.

Cook for 3 minutes with a lid. Whites should be just set completely and the yolks should be runny, bonus points for no brown edges. Season with salt and pepper.

Carefully take the sandwiches out of the broiler and plate, topping each sandwich with a fried egg. Serve immediately.