Arrange 4 slices of the sourdough bread on a flat surface. Top each with 2 to 3 slices of ham. I like to warm it up a bit first, but totally optional.Mix together half of the Gruyere cheese and the béchamel sauce. Taste for seasoning. Spread a little of the sauce on top of the ham and top each with another slice of bread.
Mix together half of the grated Gruyere cheese and the bechamel sauce. Taste for seasoning and add additional salt or white pepper if desired.
Spread a little of the sauce on top of the ham and top each with another slice of bread.
Heat a large cast iron skillet over medium heat. Add half of the remaining butter (2 TBS) and melt. Place 2 of the sandwiches in the skillet and brown on one side, about 2 minutes. Turn on the other side and brown 2 more minutes. Transfer sandwich to a baking sheet. Repeat with the remaining butter and sandwiches. Spread the remaining béchamel on top of the sandwiches and top with the other half of the Gruyere cheese.
Spread the remaining béchamel on top of the sandwiches and top with the other half of the Gruyere cheese. Sprinkle on a little more Parmesan if desired.
You can choose to top your sandwich with a fried egg or a poached egg. If frying the eggs, you can cook them in the same skillet with a little additional butter. Yolks should be runny but whites completely cooked. You can also make poached eggs in a sauce pan if preferred. While the eggs are frying or poaching, place the sandwiches under the broiler. Broil until the top becomes golden brown, 1 to 2 minutes. Watch closely! Season the eggs with salt and top each sandwich with an egg. Serve immediately.