Pozole is a hominy-based stew usually made with pork shoulder. Set bowls of shredded cabbage, avocado, radishes, chopped cilantro, and lime wedges on the table so each person can flavor their pozole as they wish. Keep plenty of tortillas and Mexican beer on hand!
Place the chiles in a heatproof medium bowl. Pour the boiling water over the chiles. Let stand until soft, about 30 minutes. Drain, reserving 1/4 cup of the liquid. Cut the chiles lengthwise in half and discard the stems and seeds. Transfer to a blender or food processor and purée with the reserved liquid. Transfer to a bowl and set aside.
Meanwhile, heat 1 teaspoon oil in a Dutch oven over medium-high heat. Add the pork and cook, turning occasionally, until browned, about 5 minutes. Transfer the pork to a plate.
Add the remaining 1 teaspoon oil, onion, and garlic to the Dutch oven. Cook over medium-high heat, stirring occasionally, until the onion is softened, about 5 minutes. Stir in the chile paste and oregano and mix well.
Return the pork to the Dutch oven. Add the broth and hominy and bring to a boil. Reduce the heat to medium low and cover. Simmer, stirring occasionally, until the flavors are blended and the pozole thickens slightly, about 1 hour. Season with salt.
To serve, ladle the pozole into soup bowls. Allow each guest to top with cabbage, avocado, radishes, and cilantro, as desired, and serve lime wedges on the side for squeezing.