Place 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 and let the potatoes cool slightly, then mash thoroughly using a potato ricer or fork/potato masher until smooth and free of lumps.
On a lightly floured work surface, combine the flour and salt in a bowl, then add the beaten egg to the mashed potatoes.
Add the flour-salt mixture to the potatoes and gently mix with your hands until the flour is moistened; transfer to the floured surface and knead briefly until the dough is smooth and slightly sticky, taking care not to overwork it.
Divide the dough into 8 equal pieces. Roll each piece into a rope about 3/4 inch in diameter on a lightly floured surface.
Cut the ropes into 3/4-inch pieces and place them on a parchment-lined baking sheet, dusting lightly with flour to prevent sticking.
Optional: To shape, press each piece against the tines of a fork or gnocchi board and roll lightly with your thumb to create ridges.
Bring a large pot of salted water to a boil. Add gnocchi in batches and cook until they float to the surface, then remove with a slotted spoon.
Serve immediately with your choice of sauce or seasoning.
