Use 2 slices of bread per sandwich. Butter one side of each slice, use about ½ tbsp butter per sandwich.
Put half the cheese on one slice, about ¼ cup per sandwich.
Place the ham onto the cheese. Shape it to cover the bread.
Pile the other half of the cheese, about ¼ cup, on top of the ham.
Place the other slice on the sandwich, buttered side down. Press down with your hand.
Frying
Beat the eggs with the salt, add the milk and heavy cream and whisk. Preheat a skillet on medium-low. Add ½ tbsp vegetable oil and ½ tbsp butter to the pan per batch (2 sandwiches).
Dip your sandwiches one by one in the egg mixture, make sure all sides are well covered. Place the sandwich immediately in the pan. This step is quite messy, wash your hand between batches.
Cook for 1½-2 minutes on one side on medium-low to your liking. Turn with a spatula and cook for another 1½-2 minutes on the other side.
Put your Monte Cristo sandwiches onto a plate. Cut in half if that's the way you like them. Serve immediately. Rinse your pan between batches.