Boil the potatoes until tender, about 20 minutes. Drain and let cool slightly.
Peel the potatoes and pass them through a potato ricer into a large bowl.
Add the flour, beaten egg, and salt to the potatoes. Mix until a dough forms.
On a floured surface, knead the dough briefly until smooth. Divide into 4 portions.
Roll each portion into a long rope, about ½ inch thick. Cut into 1-inch pieces.
Press each piece lightly with a fork to create ridges.
Bring a large pot of salted water to a boil. Add the gnocchi and cook until they float to the surface, about 2-3 minutes. Drain and serve with your favorite sauce.