Wash the potatoes, place in a saucepan, and cover with cold water. Bring to a boil, then simmer over medium heat for about 20 minutes, or until tender. Drain and leave until cool enough to peel, then put them in a bowl and mash with a potato masher or ricer. Cut the butter into small pieces and add to the potatoes. Mix with a wooden spoon.
Continuing to work the mixture, add the marjoram, parmesan, pine nuts, and a few pinches of salt. Separate the eggs and add the yolks, one at a time, mixing vigorously to amalgamate well and obtain a soft and creamy mixture. Using two spoons, form into balls the size of walnuts.
Pour the breadcrumbs onto a large plate. Whisk the egg whites together, for dipping the dumplings into.
Pour the olive oil into a frying pan, to a depth of about 2 inches (5 cm) and place over high heat. Let the oil get very hot- 375-400˚F (190-200˚C) if you have a thermometer, or otherwise drop a piece of bread into the oil, and if it sizzles right away it is ready.
Roll the cuculli first in the egg white, then in the breadcrumbs, and fry in the hot oil a few at a time for about 3 minutes, until they are golden brown on all sides. They will fluff up a little and become soft.
Drain briefly on paper towel. Serve hot, sprinkled with salt.