Combine mustard and mayonnaise; spread on each slice of bread. Make the sandwiches by covering each of 4 slices with a layer of cheese, a layer of ham and 1 more layer of cheese. Top with the remaining slices of bread.
Preheat Cuisinart™ Electric Skillet to 300°F.
Beat eggs until frothy and dip each of the sandwiches to coat evenly.
Cook sandwiches until the cheese has melted and the bread is nicely browned on each side, about 10-12 minutes.