Heat a large pot of salted water on the stove until boiling.
When water is boiling, add potatoes and cook until soft, about 20 minutes.
Remove potatoes from the water and cool in a colander. Let potatoes cool completely, then cut and remove the skin.
In a large mixing bowl, mash potatoes until they are smooth and no lumps are present. Add in flour and mix well. Next, add in the egg and continue to mix until a dough is formed. Note: You will need to use your hands!
On a floured surface, roll dough into a ball. Cover dough and rest in the refrigerator for 30 minutes.
Right before the dough is done resting, heat a large pot of salted water and bring to a boil.
Remove dough from refrigerator. Cut into four pieces. Roll out each piece into a long log, about ½ an inch thick in diameter.
Cut 1 in pieces off of the dough log, then press the edges in to form the gnocchi. You may also roll the gnocchi on a fork, or use a gnocchi roller to shape each piece.
Continue until all of the dough is used to shape gnocchi.
Cook in batches by dropping gnocchi into boiling water and cooking for 2-3 minutes. You know they are done when they float to the top of the water. Use a slotted spoon to remove the gnocchi that float to the top to a large pasta bowl. Be careful not to stir too much so the bottom gnocchi does not rise to the top.
Repeat the cooking process until all of the gnocchi is cooked. Serve with your favorite sauce, cheese, and basil.