Combine the remaining onion with 1/4 cup broth and salt. Cover and cook over medium heat until the onion is translucent, about 3 minutes. Add garlic and cook for 1 minute. Add the remaining 1 teaspoon oregano and ground chile (or chili powder) and cook for 1 minute. Add hominy and the remaining 3 1/4 cups broth, bring to a simmer and cook for 5 minutes. Add black beans and chicken, return to a simmer and cook until the chicken is no longer pink in the center, about 5 minutes.