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For the gnocchi: Bring a large pot of salted water to a boil, lower the heat and maintain at a simmer while you make the dough. In a second pot, boil the potatoes until tender. Drain right away.
Rice the potatoes and combine them on a large cutting board or countertop with the flour, pecorino, and some salt and pepper. Make a well in the flour mixture. Pour the eggs into the well. Slowly incorporate the eggs into the dry ingredients. When well combined, bring it all together in a ball and knead for one minute, until a soft dough forms. Divide the dough into quarters.
Dust the work surface and one or two baking sheets with flour. On the work surface, roll one portion of the dough into an even rope about 1-inch in diameter. Slice on the bias into gnocchi-sized dumplings using a sharp knife. Gently transfer the dumplings to the floured baking sheet. Repeat with the remaining dough.
Make an ice bath by filling a large bowl with equal parts water and ice. Add the dumplings to the pot of simmering water, working in batches to avoid crowding. When the dumplings start to float, give them another 30 seconds and then, with a slotted spoon, plunge them into the ice bath. When the dumpling have cooled, drain them well and generously oil them with olive oil. Refrigerate until ready to serve.
To assemble the dish: Heat the olive oil in a large saute pan over high heat until just beginning to smoke. Add the gnocchi to the pan, making sure they are not touching; work in batches if necessary. Let the gnocchi brown, about 1 minute, then flip them and cook for another minute. Add some of the oxtail meat and enough tomato sauce to cover the gnocchi. Cook just until the oxtail and tomato sauce is heated through, about 2 minutes, then stir in the basil and pecorino. Serve immediately, garnished with a little more pecorino. Enjoy!