Place the peeled potatoes in a saucepan and add enough cold water to cover by 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 or pass through a potato ricer until smooth; allow to cool until warm but not hot.
In a bowl, combine the flour and salt and lightly flour your work surface.
Add the beaten egg to the warm mashed potatoes, then add the flour mixture; mix with your hands until the flour is moistened.
Turn the dough out onto the floured surface and gently knead just until smooth and slightly sticky, being careful not to overwork it.
Divide the dough into 8 equal pieces and roll each into a log about 3/4 inch in diameter.
Cut the logs into 3/4-inch pieces and place them on a parchment-lined baking sheet, dusting lightly with flour as you go.
To shape (optional), press each piece lightly against the tines of a fork and run your thumb down to create ridges and a slight indentation.
Bring a large pot of salted water to a boil and cook the gnocchi in batches; they are done when they float to the surface, then remove with a slotted spoon.
Serve immediately with your desired sauce.