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TO MAKE THE SOUP: Heat the 1 tablespoon oil for about 30 seconds in a large Dutch oven or 12-quart stockpot set over medium heat.
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Salt and pepper the pork butt; sear it until browned, about 2 minutes per side.
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Remove the pot from the heat; slice the meat into 2- to 3-inch hunks and return it to the pot.
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Add the 1/2 onion, 4 cloves garlic, the 1 teaspoon salt, and the 10 cups water.
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Set the pot over high heat until the mixture boils; decrease the heat slightly to bring it down to a simmer.
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Simmer for about 1 hour, skimming off any scum that rises to the surface.
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Cover and simmer until the meat is falling-apart tender, about 1 hour more.
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Strain the meat from the broth, and return the broth to the pot.
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Skim off as much fat as possible.
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When the meat is cool enough to handle, shred it into bite-size pieces, removing and discarding as much fat as possible, and return it to the pot.
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TO MAKE THE CHILE SAUCE: While the meat simmers, place the guajillo and chipotle chiles in a large bowl and cover with boiling water.
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Use a plate or slightly smaller bowl to weigh down the chiles, keeping them completely submerged.
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Once the chiles have softened, about 30 minutes for the unusually tough guajillos, remove from the soaking water.
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Strain the soaking water, reserving 1 cup.
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Puree the chiles in a blender with 1/2 cup of the reserved soaking water, the quartered onion, and 5 cloves garlic.
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Push the pureed chile-onion mixture through a strainer to catch any seeds or large, tough bits of chile.
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Drizzle the 1 tablespoon olive oil into a skillet set over medium heat.
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Add the strained chile mixture, the remaining 1/2 cup of chile soaking water, oregano, cumin, Tabasco, and 1 teaspoon salt.
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Cook for 2 to 3 minutes, stirring occasionally.
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Stir the chile mixture into the soup; add the hominy and simmer over medium heat for about 30 minutes.
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Stir in the lime juice.
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Taste and correct the seasonings, adding salt and pepper if necessary.
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When ready to serve, set out bowls of lime wedges, sliced radishes, lettuce, green onions, cilantro, avocado, and guacamole, allowing guests to choose their own toppings.
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The soup can be made 2 days ahead, covered, and refrigerated until ready to serve.
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It can be frozen for up to 3 weeks before serving.
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Pork butt can be fatty, which makes it flavorful, but it also can lead to greasy soup.
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One of the easiest ways to remove the fat is to refrigerate the soup overnight.
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The fat will rise to the top and harden, making it easy to spoon off without sacrificing any of the flavorful broth.
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If you dont have time to refrigerate the soup, theres another way: After the straining the cooked meat from the broth, let the broth settle for about 5 minutes without stirring.
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The fat will rise to the top.
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Set a single sheet of paper towel lightly on top and let it soak up the fat.
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Discard the towel and if the broth harbors more fat, repeat.