Start the White Sauce: Place a small saucepan over medium heat. Add butter and let it melt fully. Sprinkle in the flour and stir constantly until it turns smooth and lightly golden. This takes about one minute.
Add the Dairy: Pour in the milk slowly while whisking. Follow with the cream and keep stirring. Let the sauce thicken gently until it coats the back of a spoon.
Season the Sauce: Add salt, white pepper, and nutmeg. Stir well and remove from heat. Set aside so it stays warm and smooth.
Prepare the Bread: Lay the sourdough slices on a board. Spread butter on one side of each slice. Flip them over and spread Dijon mustard on the other side.
Build the Sandwich: Place Swiss cheese on the mustard side of two slices. Add ham evenly, then another layer of Swiss cheese. Close with the remaining bread slices butter side out.
Toast the Sandwich: Heat a pan over medium heat. Place sandwiches in the pan and cook until golden on both sides. Press gently so everything warms through.
Add Sauce and Cheese: Transfer sandwiches to a baking tray. Spoon white sauce generously over the top. Sprinkle gruyere and parmesan evenly.
Finish in the Oven: Place under a hot grill until the cheese melts and bubbles. Watch closely so the top turns golden and glossy.
