In a bowl, mix the potatoes, 1/4 cup flour, cheese, and ham with a large rubber spatula until combined. Season with salt and pepper, to taste and divide the mixture into 8 equal portions. Rub your hands with a small amount of flour and form the portions into small flat disks about 1/2-inch thick. Use just enough flour to keep the potatoes from sticking to your hands, but do not coat them completely.
Heat the oil in a large nonstick skillet over medium-low heat. Shallow-fry the croquettes, turning once, until golden brown, about 4 minutes per side. Remove from the oil and drain on a paper-towel lined plate. Let the croquettes cool for about 5 minutes before serving.
Cook's Note: This recipe doubles easily if you have enough leftover potatoes.