Trim chicken breasts to approximately the same size and pound with a meat mallet, if needed, to ensure they're a uniform thickness throughout so they'll cook evenly.
In a medium pot, sauté onion in olive oil over medium-low heat for 3-4 minutes. Add garlic and sauté for 1 more minute. Add the liquid ingredients (chicken broth, tomato sauce, diced tomatoes, tomatoes and green chiles, salsa verde, and diced adobo chilis with their sauce), turn heat to high, and bring to a boil.
Turn heat down to low and add cumin, oregano, cilantro, and uncooked chicken breasts. Cook on low heat (setting 3 out of 10 on my stovetop) for 1 1/2-2 hours.
While soup is cooking, prepare rice according to package directions. Brown rice can take up to 45 minutes.
Once chicken is cooked through, remove chicken from pot and let cool for 5 minutes, then shred using two forks. While chicken is cooling, add corn, black beans, and cooked rice to pot and stir. Return chicken to soup pot and serve.