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In a large resealable plastic bag, combine the beef with the wine, mint stems and a generous pinch of salt and pepper. Seal the bag and refrigerate overnight.
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2. Preheat the oven to 325u00b0. Drain the beef, reserving the marinade; discard the mint stems. Pat the beef dry. In a large enameled cast-iron casserole, heat 2 tablespoons of the olive oil. Add half of the meat to the casserole and cook over moderately high heat, turning occasionally, until well browned all over, about 12 minutes. Transfer the meat to a plate. Brown the remaining meat over moderate heat.
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3. Return all of the meat to the casserole. Add the marinade and bring to a boil. Add the tomatoes, season with salt and pepper and bring to a boil. Cover and braise in the oven for about 2 hours and 15 minutes, until the meat is very tender.
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4. Using a slotted spoon, transfer the meat to a plate and shred with 2 forks. Boil the braising liquid until reduced to 2 1/2 cups, about 20 minutes.
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5. Meanwhile, in a large pot of salted boiling water, cook the pappardelle until al dente. Drain and return the pasta to the pot. Add the meat and the reduced braising liquid and cook over moderate heat, stirring, until the pasta is well coated with the brasato, about 2 minutes.
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6. In a small skillet, heat the remaining 2 tablespoons of olive oil until shimmering. Add the garlic and cook until lightly golden, about 1 minute. Add the mint leaves and cook for 10 seconds. Pour the garlic-mint oil over the pasta and toss. Serve in shallow bowls, passing the cheese alongside.
