Boil the potatoes in a large pot of salted water until tender, about 15 minutes. Drain and let cool slightly.
Pass the potatoes through a potato masher into a large bowl. Let cool completely.
Add the flour, egg, and salt to the potatoes. Mix until a dough forms.
Transfer the dough to a floured surface. Knead gently until smooth.
Divide the dough into 4 pieces. Roll each piece into a rope about ½ inch thick. Cut into 1-inch pieces.
Press each piece gently 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. Remove with a slotted spoon and serve with your favorite sauce.