Mornay sauce: In a medium pot over medium heat, add butter and cook until the butter is melted and starting to bubble. Add the all-purpose flour and whisk to combine the two. Allow the flour and butter to cook for 1 to 2 minutes, stirring often.
Slowly add milk while continuing to whisk. Whisking frequently will help make sure that there are no lumps in your final sauce. Once all the milk has been added, cook the sauce for 4 to 5 minutes until it starts to thicken.
Add the shredded cheese, salt, pepper, and nutmeg while continuing to whisk. Once the seasonings are fully incorporated and the sauce has thickened, you can remove the pot from the heat. If you are making your sandwich immediately you can add the lid to the pot to keep it warm while you prep everything else.
Ham and cheese sandwich: Add mustard (if using) to one slice of the bread. Top the mustard with slices of ham and then top the ham with 2 to 4 tablespoons of shredded cheese. Close the sandwich with the second piece of bread.
Griddle sandwich: In a medium-sized pan over medium-low heat, add 1/2 tablespoon of butter. When the butter is fully melted add your ham and cheese sandwich. Cook this first side until it has browned and is toasty, somewhere between 3 and 6 minutes (check often). When the first side is browned remove the sandwich from the pan briefly, add the other 1/2 tablespoon of butter to the pan, and when it has melted add the sandwich back in on top of the melted butter untoasted side down. Toast this second side for 3 or 4 minutes until it is fully toasted.
Remove the sandwich to a cooling rack while you prepare the next ingredients. Do not wash the pan. Just remove it from the heat and leave it on the stove, you will be using it again in a bit.
Turn on the broiler in your oven and allow to preheat.
Broiling: add mornay sauce to the top of the sandwich and then top the mornay with extra cheese. Make sure you go all the way to the edges with the mornay sauce so that the bread does not burn in the broiler.
Broil the sandwich in the oven for 3 to 5 minutes or until everything is melty and the cheese on top starts to turn brown and is very bubbly. Remove the sandwich to the cooling rack again to rest while you cook the egg.
Sunny-side-up egg: turn the heat under the medium-sized pan to low. With a couple of paper towels, wipe out the pan, removing all grease and residue from cooking the sandwich.
Once the pan has had 2 minutes to warm up, add 1 teaspoon of butter. When the butter is melted and bubbly, crack an egg right onto the bubbling butter.
Salt and pepper the egg and place a lid on top of the pan. Cook the egg with the lid on for 3 to 4 minutes until all the white is set.
Top the sandwich with the sunny-side-up egg, and finely chopped chives and serve.
You can make the mornay sauce in advance if you would like. When it comes time for sandwiching you can scoop out the sauce that you need for a sandwich and add a teaspoon or two of milk. Stir both together and microwave until the sauce is warmed. Then you can use as needed according to the instructions.