Put the unpeeled potatoes in a large pot. Fill the pot with enough cold water to cover the potatoes by at least 2 inches and bring to a simmer over medium-high heat. Reduce the heat to medium, partially cover the pot, and simmer the potatoes until they are completely tender and easily pierced with a skewer, 30 to 35 minutes.
Drain the potatoes, let them cool just enough that you can handle them, and then peel them. Cut them in half crosswise and pass them through a ricer into a large bowl
. Let cool until almost at room temperature, at least 20 minutes.
a work surface. In a small bowl, mix the flour with the salt. Add the egg to the potatoes and then add the flour mixture. Mix with your hands until the flour is moistened and the dough starts to clump together; the dough will still be a bit crumbly at this point. Gather the dough together and press it against the bottom of the bowl until you have a uniform mass. Transfer it to the floured surface and wash your hands.
Knead gently until the flour is fully incorporated and the dough is soft, smooth, and a little sticky, 30 seconds to 1 minute. (Don't overmix it, or the gnocchi will be tough; the dough should feel very delicate.) Move the dough to one side, making sure the surface underneath it is well floured. Cover it with a clean kitchen towel.
Cover two large rimmed baking sheets
with parchment and sprinkle lightly with flour.
Remove any lingering bits of dough from your work surface and lightly reflour the surface. Tear off a piece of dough about the size of a large lemon and put the towel back on the rest of the dough so it doesn't dry out.
With the palms of both hands, roll the dough piece on the floured surface into a rope about 3/4 inch in diameter.
With a sharp knife or a bench knife, cut the rope crosswise every 3/4 inch to make roughly 3/4-inch-square gnocchi. Arrange the gnocchi in a single layer on the parchment-covered baking sheets, making sure they don't touch. Repeat until you run out of dough, reflouring the work surface as needed. When all the gnocchi have been cut and spread out on the baking sheets, sprinkle them with a little more flour.
If you're going to use the gnocchi within 2 to 3 hours, they can sit out on the counter. For longer storage, see the make ahead tips below.
You can serve freshly made gnocchi right away or within a couple of hours, or you can freeze them for later use. Put the gnocchi in the freezer while they're still on the baking sheets and freeze until they are hard to the touch, at least one hour. Transfer them to a large zip-top bag or several smaller bags and freeze for up to two months. Cook frozen gnocchi in boiling water in two batches. Frozen gnocchi cause the temperature of the cooking water to drop, so they'll fall apart before the water returns to a boil if there are too many in the pot. Don't refrigerate fresh gnocchi for more than two or three hours, as they tend to ooze water and become soggy.
To save time, skip the fork: Classic Italian homemade gnocchi are pressed on a fork to curl them and impart the traditional ridges. To save time, I just cut them in small squares and leave them as cute little pillows. I think they look prettier, and they're a lot less fussy to make.