In a shallow bowl, lightly whisk together the milk and eggs. Season with salt and pepper; set aside.
For each sandwich, lightly butter 3 slices of bread on both sides (using 1 tablespoon of butter). Place two slices each of turkey and ham between 2 slices of bread. Top each with 2 slices of cheese, and then top with the third slice of buttered bread. Trim the crusts off, secure with toothpicks, and cut in half on the diagonal.
Melt 2 tablespoons of butter in a 12-inch nonstick skillet over medium-heat. Dip the sandwich halves in the milk mixture to coat. When the butter foams, place the sandwiches in the skillet and fry until golden brown on the bottom (about 2 minutes). Add the remaining 2 tablespoons of butter to the skillet, turn the sandwiches, and fry until browned on the other side (about 2 minutes more). Transfer to plates, sprinkle with powdered sugar, and serve with jelly.