Place the peeled potatoes in a 3-quart saucepan and cover with cold water by at least 2 inches; bring to a boil over medium-high heat and cook until fork-tender, about 20–30 minutes.
Drain the potatoes and let them cool slightly, then mash thoroughly using a potato ricer or masher until smooth and lump-free.
Lightly flour a work surface and combine the flour and salt in a bowl.
Add the beaten egg to the mashed potatoes, then add the flour mixture.
Gently mix with your hands until the flour is moistened, then turn the dough onto the floured surface and knead briefly until smooth and slightly sticky; do not overwork.
Divide the dough into 8 equal pieces and roll each into a log about ¾ inch in diameter.
Cut the logs into ¾-inch pieces and place them on a parchment-lined baking sheet dusted with flour as you work.
To shape (optional), press each piece lightly against the tines of a fork and drag with your thumb to create ridges; alternatively use a gnocchi board.
Bring a large pot of salted water to a boil and add the gnocchi in batches; they are cooked when they float to the surface, then remove with a slotted spoon.
Serve immediately with your preferred sauce or seasoning.