So for this I tried to amalgamate a few different variations.

For the sandwich
2 slices of sourdough bread per sandwich
2 slices of ham per
2 slices of Swiss per
Butter, softened

For the sauce
1 tablespoon of butter per sandwich
1 tablespoon flour per sandwich
Milk
Salt
Pepper
A bit of garlic powder
Optional - grated Swiss cheese

For the egg
Butter
1 large egg per sandwich
Salt
Pepper

Butter your pieces of sourdough, and arrange half butter side down in your medium heat pan. Arrange one slice of cheese, breaking to distribute as evenly across bread as possible, then the two slices of ham, then the second slice of cheese. Let these rest in the pan for a bit and begin to get warm before placing the rest of your bread on top.

Basically you are starting with a ham and Swiss grilled sandwich.

Once the bottom is golden brown, flip. At this point, start your broiler on high, and begin melting the butter for your sauce in the pan. Once melted and beginning to foam slightly, add the flour. Whisk until golden and beginning to bubble when resting. Add your salt, pepper, and garlic powder. Mix. Then add some milk, letting it thicken, and the repeat, until you have an overall thick cream texture. At this point add any Swiss cheese if you desire, and thin slightly with a bit more milk.

If all timed right, the sandwiches should be ready the same time the sauce is. Place sandwiches on a baking tray safe for broiling. Pour sauce over the sandwich, and place under the broiler.

Prepare your eggs in the pan you grilled the sandwiches in, I did a pseudo sunny side up by cooking with a lid over them. Keep an eye on both the eggs and the sandwiches. You remove the sandwiches from the broiler when you see the sauce on them developing brown spots from the heat.

Plate the sandwiches, and apply the egg to each.

Please note, skip the egg if you want a croque monsieur instead. That is literally the difference