In a medium saucepot, melt 4 tablespoons of butter. Add the flour and whisk until a thick paste (roux) is formed. Pour in the milk and whisk until there are no more lumps. Bring to a boil and the sauce will thicken. Add a few pinches of parmesan then season to taste with salt and pepper. Set aside on the griddle to keep warm.
Preheat Blackstone griddle to medium heat. Add the ham and cook for 3-4 minutes until heated through. Divide into 6 portions and top with Swiss cheese. Cover until the cheese is melted.
Spread butter onto the griddle and toast the bread. When the bread is toasted, spread a thin layer of Dijon mustard on each slice of bread. Build the sandwiches and top with the sauce prepared earlier.
Top the sandwiches with the parmesan cheese and cover to melt. While the sandwiches are melting, cook the 6 eggs sunny side up on the griddle. Top the sandwiches with the eggs and serve immediately.
