Melt the butter in the microwave and add it to the mashed potatoes and mix them. Add one of the beaten eggs, the ham, the cheese and the parsley. Mix well and salt and pepper to taste.
With your hands form the croquettes to the desired shape. Mine were about 3 inches long and 1 inch wide cylinders. You can also make them round or shape them like flat patties, whatever is easier for you. If small pieces of the mix want to come apart just apply light pressure to compact them again. Place the croquettes in a large plate and chill for at least 20 minutes.
Line up your three dredging and breading dishes, starting with the flour, then the remaining two beaten eggs and the Panko or bread crumbs last. Roll the croquettes on flour first, then pass them through the beaten eggs and finish by rolling them on the bread crumbs or Panko.
You can use your deep-fryer to fry them until golden brown or fry them in very hot olive oil that is deep enough to come up to one inch in a large skillet. Fry them turning them often until golden brown. Place them on paper towels to absorb any extra oil.
Serve hot with salad or vegetables.