Place the whole russet potatoes in a large pot and cover with salted water. Bring to a boil and cook until fork-tender, about 10–15 minutes depending on size. Drain and let cool until you can handle them.
Peel the cooled potatoes and mash them in a medium bowl with a potato masher until smooth and free of lumps.
Turn the mashed potatoes out onto a lightly floured board. Make a well in the center.
Add 1 cup of the flour and the salt around the potatoes, then crack the egg into the well. Mix the egg into the potato and flour until combined.
Gradually incorporate the remaining flour (up to the full 3 cups) until the dough holds together and has a soft, slightly tacky bread-dough texture. Do not overwork the dough.
Divide the dough into portions and roll each into a long log about ¾–1 inch in diameter. Cut the logs into ½-inch pieces.
To shape, roll each piece over the tines of a fork to create ridges, or leave plain if preferred.
Bring a large pot of salted water to a boil. Add gnocchi in batches and cook until they float, about 2–3 minutes. Remove with a slotted spoon and serve with your desired sauce.
