1. Heat oven to 350°F. Thinly slice 1 onion. Heat 2 tablespoons oil in large ovenproof pot over medium-high heat. Add sliced onion to pot and sauté until onion begins to soften, about 3 minutes. Add 1 tablespoon ancho chile powder and 1 tablespoon oregano; stir to coat onion. Sprinkle pork cubes with salt and add to pot. Add pork broth; bring to a boil. Cover pot and transfer to oven.
2. Braise pork until tender enough to shred easily, about 2 hours. Using slotted spoon, transfer pork to large bowl. Pour juices from pot into another large bowl. Refrigerate uncovered until cool, then cover and refrigerate overnight.
3. Discard fat from top of chilled juices; reserve juices. Chop pork into 1/2-inch cubes, discarding excess fat.
4. Thinly slice remaining 1 onion. Heat remaining 2 tablespoons oil in heavy large pot over medium-high. Sauté onion until soft, stirring often, about 6 minutes. Add remaining 1 1/2 tablespoons ancho chile powder, remaining 1 tablespoon oregano, diced chiles, garlic, and cumin; stir 30 seconds. Add chopped pork, reserved juices, and hominy. Bring to a boil; reduce heat to low and simmer 30 minutes adding more broth to thin, if desired.
5. Ladle posole into 8 bowls. Garnish with lime wedges, green onion, cilantro and avocado slices. Serve with grilled flour tortillas.