Assemble the sandwiches by spreading mustard on each slice, then topping a two pieces of bread with cheese, then ham, then more cheese. Top with the other slices of bread.
Mix together eggs and milk in a flat dish. Put a sandwich in the egg mix for 15-20 seconds per side so it can soak up a bit of the mix.
Heat a frying pan and a pat of butter while the sandwich sits in the egg mix. Preheat the oven to 300.
Cook each sandwich a few minutes on each side, adding a pat of butter before flipping so each side gets some butter.
When the bread has lightly browned on each side, put the sandwiches onto a baking sheet and then put them into the oven and bake 5-10 minutes, flipping once halfway through, so that cheese is totally melted.