Place whole potatoes in a pot covered with water to cover. Bring to a boil and simmer until they are soft, about 45-50 minutes. Peel the skins from the potatoes while they're still warm. Use a food mill or potato ricer on the finest setting and pass the potatoes through onto a clean work surface.
Make a well in the center of the potatoes and sprinkle with all of the flour. Add the egg and salt to the center of the well and use a fork to pierce the yolk. Beat the egg with the fork gradually incorporating the potato and flour. Once the egg is mixed in, bring the dough together and knead until the dough ball is dry to the touch, about 4-5 minutes.