Bring a large pot of water to a boil. Add the potatoes whole with the skins on. Boil until the potatoes are tender, then remove and let cool. Remove the skins and press through a potato ricer.
On a flat surface mix the flour and salt and form a well in the middle. Add the riced potatoes, melted butter and beaten egg to the center of the well. Use your hands to bring the ingredients together and knead to form a soft dough.
Cover the dough with plastic wrap and allow to rest for 20 minutes.
Roll out ropes of dough to ¾ inch width and cut into ½ inch pieces. Use a fork, or a a gnocchi roller if you have one, to gently press and roll the pieces to create grooves on the pieces.
Toss the pieces with a little flour to prevent them from sticking.
Cook in boiling salted water for about 3 minutes, or until they begin to float. Drain and toss with the pasta sauce of your choice.