Mash potatoes. Make sure the potatoes are well drained, then mash them using a potato ricer or masher. If you don't have one on hand, a fork and some elbow grease will get the job done. Taste for saltiness and add more salt if needed.
Combine. Combine the mashed potatoes, salt, and egg, and flour in a medium bowl. Stir using a spoon until roughly combined, then finish using your hands.
Boil water. Start boiling salted water in a large wide-bottomed pot or saucepan. Add one teaspoon of salt per 1 quart of water.
Make dumplings. Spoon about ¼ of the dough onto a well-floured work surface, then roll it into a 1 to 1 1/2-inch-thick rope using floured hands. Add more flour to the work surface if the dough starts sticking. Then cut the rope into 1 to 2-inch pieces using a lightly floured knife, and transfer them onto a well-floured surface. Cover the cut dumplings and repeat with the remaining dough.
Cook. Once the water starts boiling, lower it to a gentle simmer. Carefully transfer ¼ to ½ of the dumplings into the simmering water using lightly floured hands, and give them a gentle stir using a slotted spoon so they don't stick to the bottom. Once all dumplings float to the surface, let them cook for another 1 to 2 minutes, then remove using a slotted spoon.
Serve immediately.