Cook the potatoes, uncovered, in boiling salted water for approx. 25 mins. until soft. Drain the potatoes, peel while still hot, cut into pieces, pass through a food mill into a bowl and allow to cool slightly.
Mix the flour with all the other ingredients up to and including the pepper, mix into the potatoes along with the egg. Quickly knead into a soft, smooth dough. Cover and leave to stand at room temperature for approx. 10 mins.
Divide the dough into quarters. On a lightly floured surface, shape each piece into a roll approx. 2.5 cm in diameter and cut into pieces approx. 1 cm wide. With a little flour, shape the pieces of dough into finger-shaped dumplings that are a little thinner at the ends.
Cook the dumplings in batches in simmering salted water until they float to the surface. Remove with a slotted spoon and drain well.
Heat the clarified butter in a non-stick frying pan. Fry the dumplings for approx. 4 mins. per batch, season with salt.