Boil the potatoes until tender, about 20 minutes. Drain and let cool slightly.
Pass the potatoes through a ricer into a large bowl.
Add the flour, egg, and salt to the potatoes. Mix until a dough forms.
On a floured surface, knead the dough briefly until smooth. Divide into 4 pieces.
Roll each piece into a long rope about 1/2 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. Cook the gnocchi in batches until they float to the surface, about 2-3 minutes. Remove with a slotted spoon.