1. In a mixing bowl, whisk together the eggs and milk.
2. Assemble the sandwiches by layering ham, turkey, and cheese between two slices of bread.
3. Dip each sandwich in the egg mixture, coating both sides well.
4. Heat butter in a skillet over medium heat. Fry the sandwiches until golden brown on both sides and the cheese is melted, about 3-4 minutes per side.
5. Remove from skillet and dust with powdered sugar. Serve with jam if desired.