Bring a large pot of salted water to a boil. Add the potatoes and cook until fork tender, about 20–30 minutes, depending on their size. Drain well.
While the potatoes are still hot, peel and pass them through a potato ricer onto a clean work surface or into a large bowl. If you don't have a ricer, mash them until smooth.
Sprinkle the flour and salt over the potatoes. Make a well in the center and add the egg. Using a fork, gradually mix the egg into the potatoes and flour. Once the dough becomes too thick to stir, gently knead it by hand just until it forms a smooth, soft dough. Avoid overworking the dough.
Divide the dough into 8 portions. On a lightly floured surface, roll each portion into a rope about ½ to 1 inch thick. Cut into ½-inch pieces.
Leave the gnocchi as-is, or roll each piece over the back of a fork or a gnocchi board to create ridges that help hold the sauce.
Bring a large pot of salted water to a gentle boil. Cook the gnocchi in batches, about 15–20 pieces at a time. Once they float to the surface, cook for another 30–60 seconds before removing them with a slotted spoon.
Toss immediately with your favorite sauce and serve warm.