Whisk all of the ingredients for the egg mixture in a shallow, wide bowl. Set aside.
Make the sandwiches. Lay out your bread slices. Divide 2 teaspoons of mustard for two slices and the 2 teaspoons of raspberry jam to the other two slices. Sprinkle some salt and pepper on the mustard slices.
For each mustard covered bread slice, place the following: 1 slice of gruyere, 1 slice of Swiss cheese, 2 slices of ham, 1 slice of Swiss cheese. Lay the raspberry jam covered bread slices over the layered sandwich centers.
Preheat the oven to 400°F or 205°C and line a baking sheet with parchment paper. Set aside.
Heat a non-stick pan or cast-iron skillet. Place the tablespoon of butter in the pan and allow it to melt and bubble.
Carefully dip both sides of each sandwich in the egg mixture for 5 seconds on each side, firmly holding the layers in place.
Place the sandwiches in the pan. After 3-4 minutes, use a spatula to lightly lift the bottom of one sandwich, checking for browning. If it’s browned (like a French toast), use the bottom spatula and another spatula on top of the sandwiches to flip them quickly, firmly holding the layers in place. Using a spatula, press down on the top of the sandwiches for a few seconds.
After 3-4 minutes, check for browning again. When they’re done, transfer the sandwiches to the baking sheet and place in the oven for 5-10 minutes, checking every minute or so for fully melted cheese or your preferred doneness.
When the sandwiches are to your liking, serve each one sliced in half diagonally, preferably with a dollop of jam on the side and topped with sifted powdered sugar.