Preheat broiler. When hot, broil poblanos on a baking sheet until blackened, turning as needed (15 minutes).
Meanwhile, in a food processor, whirl garlic to mince. Cut onion in chunks and pulse with garlic until chopped. Set aside. Drain hominy and set aside.
Cut chicken into 1 inch chunks and sprinkle with salt and 1 tsp oregano. Heat oil in a 6 qt pan over high heat. Brown half the chicken lightly, stirring occasionally (5 minutes). With a slotted spoon, transfer meat to a plate. Repeat with remaining chicken.
Reduce heat to medium-high. Add onion mixture and remaining 1 tsp oregano to pan and saute until onion is softened (3 minutes). Meanwhile, in a microwave-safe bowl, microwave broth until steaming (3 minutes). Add ground chiles to pan and cook, stirring (30 seconds).
Add broth, hominy, and chicken to pan. Cover and bring to a boil, then reduce heat and simmer to blend flavors (10 minutes).
Remove stems, skins, and seeds from poblanos and discard. Chop poblanos.
Stir poblanos into posole and cook 1 minute. Ladle into bowls; top with garnishes.