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Break down carcass into smaller pieces. Simmer carcass, water, reserved leek greens, and 1/2 tsp salt in a 7- to 10-qt heavy pot, uncovered, skimming froth, 3 hours.
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Discard large bones with a slotted spoon or tongs, then strain stock through a large sieve into a large bowl (discard solids).
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If stock measures less than 10 cups, add water. If it measures more, boil until reduced.
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If using stock right away, let stand until fat rises to top, 1 to 2 minutes, then skim off and discard fat. If not, chill (covered once cool) before removing fat. (It will be easier to remove when cool or cold.)
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Cook garlic in oil in cleaned pot over medium heat, stirring, until pale golden, about 1 minute. Add onion and 1/4 tsp salt and cook, stirring occasionally, until golden, 5 to 8 minutes. Stir in chopped leeks and rosemary and cook, covered, stirring occasionally, until leeks are softened, about 5 minutes. Add carrots, celery, and 1/4 tsp salt and cook, stirring occasionally, 3 minutes. Stir in tomatoes and stock and simmer, uncovered, stirring occasionally, until vegetables are tender, about 20 minutes. Season with salt and pepper.
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Stir in pasta and briskly simmer soup, stirring occasionally, until pasta is al dente. Add spinach and stir until wilted.
