Place the whole russet potatoes in a large pot, cover with salted water, and simmer until a fork easily pierces them, about 10–15 minutes depending on size; drain and let cool until you can handle them.
Remove the skins and mash the warm potatoes in a mixing bowl with a potato masher until very smooth and free of lumps.
Turn the mashed potatoes out onto a lightly floured work surface and let cool enough so they won’t cook the egg when added.
Make a well in the potatoes and sprinkle 1 to 1½ cups of the flour and 1½ teaspoons salt around the mound; crack the egg into the well.
Gently mix the egg into the potatoes, gradually incorporating the flour, then add the remaining flour as needed until a soft, slightly tacky dough forms—similar to bread dough but not sticky.
Divide the dough into portions and roll each into long ropes about ¾ inch in diameter; cut the ropes into ½-inch pieces.
Optional: roll each piece over the tines of a fork to create ridges, pressing gently to form the classic gnocchi shape.
Bring a large pot of salted water to a boil, add the gnocchi in batches, and cook until they float to the surface, about 2–3 minutes; remove with a slotted spoon.