Make the chicken and stock: in a large Stock Pot, combine the chicken, onion, carrots, celery, garlic, peppercorns, parsley, and thyme. Cover with 6 quarts cold water and bring to a boil over high heat. Reduce the heat to a light to moderate simmer and cook for about one hour.

Transfer the chicken to a cutting board and let cool. Strain the broth through a fine-mesh sieve set over a large bowl and discard the solids. Measure 4 quarts (16 cups) of the stock and set aside (reserve the remaining stock for another use). When the chicken is cool enough to handle, shred the meat and set aside.

Start the posole: In a large, dry Stainless Clad or Carbon Steel Frying Pan, gently toast the chiles over medium heat until they darken in spots and begin to smell fragrant, about 2 minutes—move them around the pan to promote even toasting. Transfer them to a small bowl and cover with hot water to soften. Let sit for about 20 minutes.

Pour 1 can of hominy, including the liquid in the can, into a food processor or blender and blend until it forms a puree. Set aside. Drain the other can of hominy in a fine-mesh sieve, discarding the liquid. Rinse the hominy and set aside separately from the puree.

Drain the chiles, reserving the water, and remove the stems and seeds. Transfer them to a food processor, add ½ cup of the soaking water, and puree until a paste forms. Set aside.

In a large Dutch Oven, heat the oil over medium heat until shimmering-hot. Add the onion and cook, stirring, until it softens and gets slightly translucent, about 8 minutes. Add the garlic and cook, stirring, until you can really smell it, 1–2 minutes. Add the reserved chile paste, salt, cumin, and oregano and cook, stirring, for another 2 minutes to toast the spices. Add hominy puree, drained whole hominy, pulled chicken meat and as much additional reserved stock as you can to fill your Dutch oven (without overflowing when it comes to a boil), anywhere from 10–16 cups. Increase the heat to high to bring to a boil, then reduce to a simmer and cook for 1 hour.

To serve, ladle the soup into bowls and garnish with some cabbage, onion, cilantro, and a squeeze of lime.