Place potatoes in a large saucepan and cover with 2 inches of salted water. Bring to a boil over high heat; reduce heat to medium-high and simmer, undisturbed, until tender but still firm, 25 to 30 minutes. Drain potatoes and place back in the pot to dry and cool slightly, 5 to 10 minutes.
As soon as potatoes are cool enough to handle, peel potatoes by rubbing skins off with your hands. Pass peeled potatoes through a potato ricer into a large bowl (alternatively, mash until very smooth.) Let cool slightly, about 10 minutes.
Fold in flour, egg, and salt until flour is incorporated and dough starts to hold together. Transfer dough to a lightly floured surface and gently knead once or twice until mostly smooth (do not overwork).
Divide dough into 4 even portions. Roll and shape each portions of the dough into long "snakes," about ¾-inch in diameter.
Cut snakes into about 1/2-inch long pieces. (Optional: Roll each piece against the tines of a fork or a gnocchi board to make light ridges.) Transfer shaped gnocchi to a single layer on a parchment-lined baking sheet dusted with flour.
Bring a large pot of lightly salted water to a boil over high. Working in batches, drop in gnocchi and cook for 3 to 4 minutes or until gnocchi have risen to the top; transfer cooked gnocchi with a spider or slotted spoon to desired sauce.
