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1
Put the chicken, the beef bones if you're using them, 3 of the onion quarters and the garlic in a large pot. Add water just to cover (about 10 cups) and bring to a boil. Reduce the heat so the liquid bubbles gently. Cook, skimming the foam off the surface every now and then, until the chicken is very tender, 45 minutes to 1 hour.
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2
Meanwhile, put the vegetable oil in a large skillet over medium heat. When the oil is hot but not smoking, fry 2 of the tortillas (one at a time if necessary), turning once, until crisp and golden, 2 to 3 minutes per tortilla. Drain on paper towels. Cut the 4 remaining tortillas into strips, add them to the skillet and fry, stirring to separate them, until crisp and golden, another 2 to 3 minutes. Drain on paper towels and sprinkle with salt while they're still warm.
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3
When the chicken is tender, transfer it to a plate or cutting board with tongs or a slotted spoon (or put it in the fridge or freezer so it cools faster). When it's cool enough to handle, shred the meat with your fingers, discarding the bones and the skin. (If you used beef, discard it or save it for another use.)
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4
While the chicken is cooling, strain the stock and discard the solids. Peel the remaining quarter of an onion and put it in a blender with the chipotle, cumin, oregano, roasted tomatoes, 1/4 cup of the cilantro and a sprinkle of salt. Crumble in the two whole fried tortillas and add enough stock to fill the blender a little more than halfway. Puree until the mixture is as smooth as possible.
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5
Pour the puree and remaining stock back into the pot and bring to a boil. Reduce the heat so the mixture bubbles gently and cook for 5 to 10 minutes. Stir in the shredded chicken, hominy and black beans. Taste and add more salt if necessary. Divide the avocados, the cheese and the remaining 1/4 cup cilantro among 4 to 6 bowls. Ladle the soup into the bowls and garnish with the fried tortilla strips. Serve immediately with lime wedges if you like.
