3 large baking potatoes (about 2 pounds), scrubbed
1/4 cup plus 2 tablespoons cornstarch
6 tablespoons all-purpose flour
1/2 cup freshly grated Parmigiano-Reggiano cheese (2 ounces)
1 large egg, beaten
Canola oil, for frying
Ketchup, for serving
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 towellined platter and turn until dry. Repeat with the remaining gnocchi.
In a large saucepan, heat 2 inches of oil to 375°. 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 towellined plate and sprinkle with salt. Repeat to make the remaining "Tater Tots." Serve with ketchup.