Prepare the Egg Mixture: In a shallow baking dish, whisk together the milk, eggs, vanilla extract, ground cinnamon, and nutmeg. Set aside.

Assemble the Sandwich: Lay out the brioche slices on 4 slices, layer 1 slice of Swiss cheese, followed by 2 slices each of the double-smoked ham and turkey. On the other 4 bread slices, spread 1 teaspoon (or more) of Dijon mustard and top with another slice of cheese. Bring the opposing bread sliced together to form 4 sandwiches total.

Preheat the Griddle: Heat your Blackstone griddle to medium heat. Add a good amount of butter to the heated surface, ensuring it's evenly coated and ready to cook your sandwiches.

Egg Dip: Lightly press the Monte Cristo sandwiches down to keep them from falling apart during dipping. Carefully dip each sandwich into the egg mixture, coating both sides. Do not oversoak the bread.

Griddle: Place the dipped sandwiches on the Blackstone griddle. Cook the sandwiches for 3-4 minutes on each side until they are golden brown and the cheese is melted.

Garnish and Serve: Transfer Monte Cristo sandwiches to a cutting board and let them rest for 1-2 minutes. Lightly dust with powdered sugar and serve sandwiches with raspberry jam (or berry jam of choice) for dipping.