Pound out each piece of pork with a meat mallet until 1/8 inch thick. Season with salt and pepper on both sides and top each with a single layer of ham then cheese, making sure the ham and cheese do not go all the way to the edges of the pork. Roll each up into long thin rolls (if the rolls are too big, you may need to cut them into smaller pieces so they are no more than 2 inches thick).
Place the flour in a shallow dish, beat the eggs with a splash of water in another, and spread the panko in a third.
Dredge all sides of each rolled pork in flour and shake off the excess. Coat in the beaten egg then dredge in the panko.
Add enough oil to a large skillet to come up 1/4 inch. Heat over medium heat until hot. Add the flamenquines in batches and fry, turning occasionally, until golden brown all over, about 8 minutes.
Drain on paper towels and season with some more salt. Cut each flamenquine on the bias into beautiful slices or pieces and serve.