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1
Heat 2 teaspoons vegetable oil in a large, heavy pot over high heat; brown beef shank slice in hot oil until seared and browned on both sides, about 3 minutes per side. Add onion, carrot, celery, water, bay leaf, 3 cloves peeled garlic, and black peppercorns; bring mixture to a simmer. Reduce heat to low and simmer until meat falls off the bone and stock turns brown in color, 4 to 5 hours. Add more water if beef stock level gets below about 5 cups.
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2
Place beef short ribs into freezer until firm and very cold, about 15 minutes. Cut meat into cubes and return to freezer until almost frozen, about 30 more minutes.
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3
Whisk egg with parsley, 2 cloves garlic, and cream in a bowl. Stir in bread crumbs and 1/4 cup Parmigiano-Reggiano cheese.
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4
Place partially-frozen beef cubes into food processor and pulse with with 1/3 teaspoon salt and 1/8 teaspoon black pepper until beef is coarsely ground, 5 to 10 pulses. Pulse in bread crumb mixture until mixture is finely ground, about 5 more pulses. Transfer meat to a bowl, cover with plastic wrap, and refrigerate.
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5
Strain beef stock into a large soup pot, discarding beef and vegetables. Mix chicken broth into beef stock and whisk in tomato paste. Bring broth mixture to a simmer and season with salt and black pepper to taste. Stir kale into broth; simmer kale until tender, about 15 minutes.
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6
Scoop 2-teaspoon portions of the meat mixture and roll into meatballs.
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7
Pour pastina into the simmering soup, cook for 5 minutes, and gently add meatballs. Simmer until meatballs are cooked through and tender, about 15 minutes. Serve in bowls garnished with a pinch of black pepper or a sprinkle of shredded Parmigiano-Reggiano cheese.
