Preheat broiler. When hot, broil poblanos on a baking sheet until blackened, turning as needed, about 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; set aside.
Cut chicken into 1- to 1 1/2-in. chunks and sprinkle with salt and 1 tsp. oregano. Heat oil in a 5- to 6-qt. pan over high heat. Brown half the chicken lightly, stirring occasionally, about 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 sauté until onion is softened, 3 minutes. Meanwhile, in a microwave-safe bowl, microwave broth until steaming, about 3 minutes. Add ground chiles to pan and cook, stirring, about 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.
*Find Mexican oregano at well-stocked grocery stores, along with ground red New Mexico chiles.
Note: Nutritional analysis is per serving.