Step 1

In a large enameled cast-iron casserole, heat 2 tablespoons of the olive oil. Season the pork with salt and pepper, add to the casserole and cook over moderately high heat, stirring twice, until browned, about 8 minutes. Add 1 piece of the quartered onion, the cloves, the 1 tablespoon of oregano, the bay leaves and 6 cups of the beef broth and bring to a boil. Cover and simmer over moderately low heat until the pork is tender, about 40 minutes. Discard the bay leaves.

Step 2

Meanwhile, heat a large cast-iron skillet. In a large bowl, toss the tomatillos, tomatoes, garlic and remaining 3 pieces of onion with 1 tablespoon of the olive oil; season with salt and pepper. Add the vegetables to the skillet and cook over moderately high heat until blistered in spots, about 10 minutes. Transfer the vegetables to a blender.

Step 3

In the same skillet, toast the chiles over moderately high heat until slightly darkened, about 2 minutes. Add the remaining 2 cups of beef broth and bring to a simmer; cover the skillet, remove from the heat and let the chiles soften for 5 minutes. Add the chiles and broth to the blender with the vegetables and puree until smooth. Strain the vegetable-chile puree into a bowl. Wipe out the skillet.

Step 4

Heat the remaining 1 tablespoon of olive oil in the skillet. Add the chorizo and cook over moderate heat, breaking it up with a spoon, until browned, about 5 minutes. Add the chorizo, hominy and vegetable-chile puree to the casserole and bring the pozole to a simmer. Season with salt and pepper. Garnish the pozole with radishes, onion, cilantro and oregano and serve with lime wedges.