Spread all the bread slices, 1 side only, with the butter and mustard. Add the 2 slices of ham and 2 slices of cheese to each of 4 bread slices. Cover with remaining bread to make a traditional sandwich.
In a large shallow bowl add the beaten eggs and the whole milk. Season with salt and pepper, to taste, and combine well. Dip each sandwich into the egg mixture making sure you cover the whole sandwich, including the sides. Place on a large platter until ready to cook. Heat the remainder of the softened butter in a large skillet. When the butter has melted and the pan is very hot, add the sandwiches and cook turning once, until golden brown, about 4 to 6 minutes. Transfer to serving plates and serve immediately.