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Prepare the leeks. Trim and thoroughly wash the leeks, taking care to remove grit between layers. Cut the mostly white stalks into pieces about 7 cm / 3 inches long. Reserve 200–250 g of the inner leek hearts for the filling and the remaining green portions for lining the pot.
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Make leek cylinders. Slit each leek section lengthwise toward its center and gently remove the smaller inner leaves. Keep 2–3 outer layers together as a cylinder for stuffing. Reserve the removed leaves for lining the pot.
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Sauté the aromatics. Heat some of the measured olive oil in a skillet. Sauté the chopped leek hearts and then the onion until softened. Remove from heat.
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Mix the filling. Add the uncooked rice, dried thyme, ground allspice and raw ground beef to the sautéed vegetables. Season with salt and pepper, mix thoroughly, then stir in the beaten egg.
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Stuff the leeks. Spoon filling into each prepared leek cylinder. Use an additional slit leek layer to cover the opening and help keep the filling in place.
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Arrange in the pot. Line the bottom of a wide, heavy pot with reserved leek greens and inner leaves. Arrange stuffed leeks snugly in layers. Pour in the remaining olive oil, diluted tomato paste, whole allspice berries and enough broth to come about three-quarters of the way up the leeks, without completely submerging them.
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Simmer. Cover the stuffed leeks with leftover leek leaves and place a heat-safe inverted plate over them to hold them in place. Cover the pot and simmer gently over low to medium-low heat for about 45 minutes, until the rice and beef are fully cooked and the leeks are tender. Check liquid occasionally; leeks release their own moisture.
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Finish sauce. Remove the plate carefully. Stir the cornstarch slurry into the simmering pot liquid and cook briefly until slightly thickened. Serve warm with some of the sauce.