Lay out 4 slices of bread. Top each with a slice of cheese, then ham, and turkey.
Cover with another slice of bread to form sandwiches.
In a shallow bowl, whisk together eggs, milk, salt, and pepper.
Dip each sandwich into the egg mixture, coating both sides well.
Melt butter in a skillet over medium heat.
Cook sandwiches 3–4 minutes per side until golden brown and cheese is melted.
Dust with powdered sugar and serve with raspberry jam on the side.