In a large, enameled cast-iron casserole, bring the chicken stock and water to a boil. Add the chicken breasts, skin side down, cover and simmer over very low heat until they're tender and cooked through, about 25 minutes. Transfer the chicken breasts to a plate and shred the meat; discard the bones and skin. Skim any fat from the cooking liquid and reserve.

In a blender, combine the halved tomatillos with the quartered onion, poblanos and jalapenos, smashed garlic, chopped cilantro, and oregano. Pulse until coarsely chopped, scraping down the side. With the machine on, add 1 cup of the cooking liquid and puree until smooth. Season the tomatillo puree with salt and pepper.

In a large deep skillet, heat the vegetable oil until shimmering. Add the tomatillo puree and cook over moderate heat, stirring occasionally, until the sauce turns a deep green, about 12 minutes.

Pour the green sauce into the cooking liquid in the casserole. Add the hominy and bring to a simmer over moderate heat. Add the shredded chicken to the stew, season with salt and pepper and cook just until heated through. Serve the pozole in deep bowls, passing the lettuce, radishes, onion, avocado, sour cream, tortilla chips, and lime wedges at the table.

The pozole verde can be prepared through Step 3 and refrigerated, covered, overnight.