1. Put the potatoes in a large pot, add water to cover, and bring to a boil. Reduce the heat and cook at a low boil until the potatoes are tender, about 45 minutes; drain.
2. While they are still warm, peel the potatoes, then pass them through a vegetable mill onto a clean work surface.
3. Bring 6 quarts of water to a boil in a large pot, and add 1 tablespoon salt. Set up an ice bath nearby. Make a well in the center of the potatoes, and sprinkle them all over with the flour. Break the egg into the center of the well, add the salt, and, using a fork, blend the egg and salt together. Using the fork, begin to incorporate the flour and potatoes as if you were making pasta. Once the dough begins to come together, begin kneading it gently until it forms a ball. Knead gently for another 4 minutes, or until the dough is dry to the touch.
4. Divide the dough into 6 balls. Roll one ball into a rope 3/4 inch in diameter, and cut it into 1-inch pieces. Roll each piece down the back of a fork to create the characteristic ridges. Drop the gnocchi into the boiling water and cook until they float to the surface, about 1 minute. Use a slotted spoon to transfer the gnocchi to the ice bath. Repeat with the remaining dough, replenishing the ice as necessary.
5. When all the gnocchi have been cooked and cooled in the ice bath, drain them and transfer to a bowl. Toss with the oil. The gnocchi can be stored, covered, in the refrigerator for up to 48 hours.