Mix mayo and mustard together in a small bowl.
Spread onto 4 bread slices.
Add cheese and ham to each bread slice.
Place the other slice of bread on top of the ham.
Meanwhile, whisk eggs and milk in a pie plate until well blended. Place butter in a skillet and melt.
Dip each sandwich in egg mixture (make sure all sides are coated), then place in buttered pan. Fry until golden brown on both sides (a few minutes on each side).