1) Lay out three slices of sandwich bread. Split grated cheese between two slices of bread. (You can sub smoked gouda if you want). Top cheese with sliced ham.
2) Lightly spread mayo on BOTH sides of the third piece of bread. Place the mayo piece of bread on top of one of the two sides. Then flip the other ham/cheese side on top so the mayo piece of bread is in between the two sides, creating a two layer sandwich. The cheese should be on the outsides of the sandwich.
3) Slices the ends off the sandwich and press it with some weight to really compact the layers. You can just cover it with a flat plate or cutting board and press on it gently for a few seconds.
4) Whisk together the egg, milk, paprika, and a pinch of salt and pepper.
5) Heat butter in a small skillet over medium heat.
6) When butter is melted, dredge pressed sandwich in the egg mixture, coating all sides. Immediately move sandwich to the skillet. Cook for 3-4 minutes per side until sandwich is golden brown. Also, upend the sandwich to cook on the ends for about 15 seconds per side.
Sandwich is done cooking when it is a dark golden brown and cheese is completely melted. Cut sandwich in half and serve immediately!