Spread 2 bread slices with jam and 2 bread slices with mustard. Top each slice of jam-topped bread with 1/4 cup cheese, 1-1/2 ounces ham, 1-1/2 ounces turkey and 1/4 cup cheese. Top with remaining bread, mustard side down.
In a shallow bowl, whisk eggs, milk, vanilla, cinnamon and nutmeg until blended. In a large skillet, heat 2 tablespoons butter over medium heat. Dip both sides of sandwiches in egg mixture. Place in skillet; toast until golden brown and cheese is melted, 2-3 minutes on each side. Dust with confectioners' sugar and serve with additional jam.