Mix the mustard and mayonnaise in a small bowl. Lay out 3 slices of bread and brush each with about 1 teaspoon of the mustard mixture. Top each piece of bread with 1 slice each of cheese, turkey and ham. Cover with another slice of bread and brush with another teaspoon of the mustard mixture; top each sandwich with another slice each of cheese, turkey and ham. Brush the remaining 3 bread slices with the mustard mixture and place mustard-side down on top of the sandwiches.
Trim the crusts, then wrap the sandwiches tightly with plastic wrap. Chill in the refrigerator at least 30 minutes and up to 6 hours. (Wrapping the sandwiches helps keep them compact.)
Whisk the eggnog and nutmeg in a shallow bowl. Unwrap the sandwiches and dip in the eggnog, turning to coat both sides. Melt half of the butter in a large nonstick skillet over medium heat. Add the sandwiches and cook until the undersides are golden, 3 to 4 minutes. Add the remaining butter, flip the sandwiches and cook until golden on the other side, about 3 more minutes.
Cut the sandwiches into quarters. Dust with confectioners' sugar, if desired, and serve with cranberry sauce.