Put the potatoes in a large saucepan and cover with 4 inches of cold water. Bring to a boil and simmer over moderately low heat until tender, about 40 minutes. Drain and let cool slightly. Cut the potatoes into large chunks and press them through a ricer into a large bowl. Discard the skins. Let the potatoes cool completely, about 1 hour.

Stir the cornstarch, flour, cheese, and 2 teaspoons of salt into the potatoes. Add the egg and knead gently until a soft dough forms. Wrap the dough in plastic and let stand at room temperature for 15 minutes.

Bring a very large pot of water to a boil and fill a large bowl with ice water. Divide the dough into 4 pieces and roll each piece into a 16-inch rope, about 1 inch thick. Cut each rope into 16 pieces. Add one-fourth of the gnocchi to the boiling water and cook, stirring once or twice, until they rise to the surface and then float for 20 seconds. Using a slotted spoon, transfer the gnocchi to the ice water to cool, then spread the gnocchi on a paper towel-lined platter and turn until dry. Repeat with the remaining gnocchi.

In a large saucepan, heat 2 inches of oil to 375 degrees. Carefully add one-fourth of the gnocchi to the hot oil and cook, stirring gently once or twice, until golden and crisp, 4 to 5 minutes. Transfer the "Tater Tots" to a paper towel-lined plate and sprinkle with salt. Repeat to make the remaining "Tater Tots." Serve with ketchup.