Trim away large pieces of fat from the pork, cut the roast into 1-inch cubes, and place it in a large stock pot with enough water to cover it by about 2 inches, using 8-10 cups water. Bring the water to a boil and skim away the foam that collects on the surface.

Add the quartered onion, 5 whole garlic cloves, bay leaves, 2 teaspoons salt, and chicken bouillon to the pot, then reduce the heat to a gentle simmer for 1 ½ hours. The broth should bubble softly while the pork begins to soften and smell deeply savory.

Remove the stems, seeds, and veins from the dried chiles, then place them in a small pot with enough water to cover them. Boil for 15 minutes, until the chile skins are softened and flexible.

Transfer the softened chiles to a blender with a slotted spoon, then add 1 cup of the chile cooking liquid and the remaining 5 peeled garlic cloves. Blend until the mixture becomes a smooth, thick red puree.

Discard the chile cooking water, heat 2 tablespoons oil in the small pot over medium heat, and strain the blended chile puree into the oil through a fine mesh strainer. Cook over medium-low heat for 10 minutes, stirring often, until the sauce smells mellow and concentrated.

Remove the bay leaves, onion, garlic cloves, and neck bones (if used) from the pork pot, then stir in the chile sauce, tomato sauce, oregano, cumin, white vinegar, and hominy. Bring the pozole to a slow boil, then simmer for 30 minutes, until the pork is very tender and the broth is brick red.

Taste the broth and adjust with salt, chicken bouillon paste, or oregano as needed. The finished broth should taste savory, lightly tangy, and balanced rather than flat or harsh.

Ladle the hot pozole into bowls and add shredded cabbage, diced onion, cilantro, and lime juice, with avocado and radishes if desired. The fresh toppings should stay crisp against the rich, steaming broth.