Put about two inches of water in a large pot fitted with a steamer basket and place pot over high heat. Add potatoes, cover, and steam until fork tender, about 45 minutes. Drain well, then peel: Holding each potato with a thick, dry kitchen towel (they will be too hot to hold directly), remove the skin with a paring knife. Immediately pass the potatoes through a ricer. Spread out on a baking sheet and let cool completely.
Turn out potatoes onto a work surface, then mix egg with one tablespoon salt and pour over the potatoes and add flour in 1/2 cup increments, up to 1 1/2 cups. Start to work the mixture with your hands and bring it together to form a dough; then gently knead 4 to 5 minutes, adding more flour (up to 1/2 cup total) as necessary to keep the dough from sticking, until dough is smooth and elastic. (To check, pinch off a piece and roll into a rope; it should not break apart).
Line a rimmed baking sheet with a dry clean kitchen towel (or parchment paper) and sprinkle liberally with flour. Use a bench scraper to divide dough into four to six pieces. Begin gently rolling each piece under the palms of your hands into a rope, then continue rolling until it is Â½ inch in diameter. Use the bench scraper to cut ropes crosswise into 1-inch pieces. Roll a cut side of each dumpling against the tines of a fork with your thumb (each piece will have ridges on one side and an indentation on the other). Set gnocchi in a single layer on the prepared baking sheet. If not cooking immediately, cover with plastic wrap and refrigerate up to 2 hours.
Bring a large pot of water to a boil, then add a generous amount of salt. Working in small batches, add gnocchi and cook until they float to the top, about 2 minutes. Remove gnocchi with a slotted spoon and gently shake off excess water before placing in a large pasta bowl. (If not serving immediately, plunge gnocchi into an ice-water bath to stop the cooking. Once completely cool, drain thoroughly, toss with a little extra-virgin olive oil, and refrigerate in a covered container for up to 3 hours. Reheat slowly on the stove, with the serving sauce or melted butter. Freezing gnocchi is not recommended.) Toss with pesto while boiling remaining gnocchi.
Once the last batch has been boiled, drained, and tossed with pesto, divide gnocchi among bowls and serve at once, garnished with basil.