Put the potatoes in a large saucepan over medium heat with enough water to cover by an inch. Bring to a simmer and cook until tender all the way through when pierced with a knife, about 20 minutes. Drain.
When the potatoes are still hot but you are able to handle them, peel off the skins with a paring knife and press the warm potatoes through a ricer in a flat layer on a working surface. Let cool completely.
Once the potatoes are cool, sprinkle with 1 teaspoon kosher salt. Drizzle the beaten egg over the potatoes. Sprinkle with 1 cup of the flour and use your hands (or preferably a bench scraper, if you have one) to mix and knead the dough together. Add 1/4 to 1/2 cup more flour as needed until the dough holds together and is only slightly sticky. (A good way to tell if the dough is the right texture is to cut the dough ball in half--the interior should look like cookie dough peppered with small holes throughout.)
To form the gnocchi, break off a piece of dough and roll into a log about 3/4 inch thick, keeping your hands and counter well floured as you work. Cut crosswise into 3/4-inch segments. Give each piece a quick single roll between your palms to round them slightly. Roll each piece off the back of the floured tines of a fork, pressing with your thumb as you roll down to make a small indentation on one side and ridges from the fork on the other side. Place on a floured baking sheet.
Bring a pot of water to a boil and salt generously. Boil the gnocchi until cooked, 2 to 4 minutes. Drain and serve tossed in a sauce of your choosing. Garnish with Parmesan and basil.
Extra gnocchi can be frozen. First, freeze them until solid right on the floured baking sheet. Then pack them in plastic ziptop bags for storage.