Thick and crusty bread works best. You want bread that can soak up the Mornay sauce, but still have some structure. Thin sandwich bread will work, but it won’t be as hardy.
Toasting the bread before spreading the mustard on one side is an extra step, but it helps the sandwich hold up through the process.
Thinly sliced deli ham works very well; get an assortment of different types of ham, if you like. Don’t think you have to adhere to the three or four slices per sandwich. Double it if you want because this is not a low-cal sandwich. Thin slices of country ham will work, too, but make sure to cook them beforehand.
French Gruyere is traditional (this is a French sandwich); Swiss Gruyere, Emmental or Swiss cheese also work. Keep in mind that you need cheese for the sauce as well as the sandwich, and extra cheese is better than not enough.
If I am going to the deli anyway for ham, I like to get sliced Swiss cheese for assembling the sandwiches and some nice Gruyere to grate for the Mornay sauce.
Melting cheese in a pan is not a cheese sauce; you will have a big gloppy mess of separated cheese that you won’t be able to pour or spread on anything. The base to any dairy sauce or soup is called a béchamel — milk/dairy with a white roux (heated fat and flour as a thickening agent).
The process can be achieved in a few different ways. Cold milk can be whisked into hot roux or cold roux can be whisked to hot milk. Keeping the temperatures opposite helps the roux emulsify and incorporate without leaving lumps of flour. The ratio is about 1 pound of roux per gallon of liquid, which breaks down to about 1 ounce of roux (1 tablespoon flour, 1 tablespoon fat) per cup. After mixing the roux and the milk, slowly bring to a simmer for 10 to 15 minutes to cook away the raw-flour taste, stirring often. Add a small amount of nutmeg and cheese, season with salt and pepper, and use as desired.
You can make Mornay in large batches; cool slightly and split into small containers or zip-close bags and freeze. When you feel like having a sandwich, pull out a bag, thaw and use.
If you are making one or two sandwiches, grill them start to finish in the pan as if you were making a grilled cheese. If you are making more than two, assemble all the sandwiches and bake them all at once in a 225- to 250-degree oven until the cheese is melted and ham is hot. You can then brown both sides of the sandwiches in a sauté pan with butter. The sandwiches can sit for a bit after browning since they are going into the oven again before serving.
When you are ready to serve, preheat the broiler to high or set the oven to 500 degrees. Slather the Mornay sauce over the top of the sandwiches and place into the heat for 3 to 5 minutes until the top has started to brown slightly and bubble. If desired, cook an over-easy egg to place on top of the sandwich after broiling.