In a shallow dish that will be large enough to dip your sandwich in, whisk the egg with some salt and pepper. Set aside.
Melt 1 tablespoon of butter in a skillet over medium-high heat. Spread the mayonnaise and mustard onto one slice of bread. On the other slice, layer the ham and cheese. Place the slices together to enclose the ingredients.
Carefully and quickly, dip each side in the beaten egg, and place in the hot buttered skillet. Cook until golden brown on the bottom, then melt the remaining butter in the skillet. Flip the sandwich, and cook until browned on the other side.