Place whole, unpeeled potatoes in a large pot of lightly salted water and boil until tender when pierced, about 20–30 minutes depending on size.
Drain and let the potatoes cool until they are warm but safe to handle; peel and discard the skins.
Press the warm potatoes through a potato ricer or food mill into a large bowl to create a light, lump-free texture; do not mash with a fork.
Add the flour, salt, and the beaten egg to the riced potatoes. Gently combine with your hands until a smooth, soft dough forms, taking care not to overwork it; add a little more flour only if the dough is sticky.
Divide the dough into four pieces. On a lightly floured surface, roll each piece into a rope about 3/4 inch in diameter and cut into 3/4-inch pieces.
Shape each piece by pressing the center with your thumb and rolling it down a floured gnocchi board or the tines of a fork to create ridges and a curled shape. Toss finished gnocchi lightly in flour to prevent sticking and spread in a single layer on a baking sheet to air dry about 30 minutes (or freeze 2–3 hours if desired).
Bring a large pot of lightly salted water to a rapid boil. Add gnocchi in batches, stir once, and cook until they float to the surface, about 2–3 minutes.
Use a slotted spoon to remove the cooked gnocchi as they rise, and immediately toss with your sauce of choice, adding a little reserved cooking water if needed to loosen the sauce.