Place the whole, unpeeled potatoes in a large pot of cold water and bring to a boil; cook until fork-tender, about 20–30 minutes.
Drain and let cool slightly until safe to handle, then cut each potato lengthwise and scoop the flesh into a ricer fitted with the finest disk; rice directly onto a clean work surface.
Spread the riced potato into an even layer, sprinkle with a pinch of sea salt, then add about half the flour and half the beaten egg.
Gently knead the mixture, incorporating the flour and egg; add the remaining flour and egg as needed, kneading until the dough is soft but holds together without becoming sticky.
Cut off a small portion of dough and roll it on a lightly floured surface into a rope about 1/2 inch (1.2 cm) thick.
Cut the rope into 1-inch (2.5 cm) pieces, then press each piece with your index finger and roll on a gnocchi board or the tines of a fork to form ridges.
Place formed gnocchi on a well-floured baking tray in a single layer and repeat with remaining dough.
Bring a large pot of lightly salted water to a gentle boil and cook the gnocchi in batches for 1–2 minutes, until they rise to the surface.
Use a slotted spoon to transfer the cooked gnocchi to your sauce or a serving dish, then toss gently and serve.
If not cooking immediately, freeze the gnocchi in a single layer on a baking sheet, then transfer to freezer bags and cook from frozen when needed.