Lay out the 4 slices of brioche bread on a clean work surface. Spread a thin layer of dijon mustard on one side of each slice. On 2 of the slices, layer 2 slices of swiss cheese, 2 slices of smoked turkey, and 2 slices of black forest ham on each.
Top each stacked sandwich half with the remaining bread slices, mustard-side down. Press down gently to compact the sandwiches.
In a shallow dish, whisk together the eggs, whole milk, and ground nutmeg until smooth. Carefully dip each assembled sandwich into the egg mixture, letting it soak for about 10 seconds per side. You want the bread coated but not soggy.
Preheat the air fryer to 370 degrees F for 2 minutes. Spray the air fryer basket generously with nonstick cooking spray. Place the sandwiches in the basket and spray the tops lightly with cooking spray as well.
Air fry at 370 degrees F for 4 to 5 minutes on the first side, then carefully flip the sandwiches. Cook for another 3 to 4 minutes, until both sides are golden brown and the cheese is melted through.
Remove the sandwiches and let them rest for 1 minute. Cut each sandwich in half diagonally, dust generously with powdered sugar, and serve with a small bowl of raspberry jam or preserves for dipping.