Sauté the vegetables. Heat olive oil in a large pot over medium heat. Add diced onion and poblano peppers, cooking for 8-10 minutes until soft.
Toast the spices. Stir in the minced garlic, cumin, oregano, chili powder, salt, and pepper. Cook for 1 minute until fragrant.
Simmer the soup. Pour in the chicken broth, tomatoes with juices, and black beans. Submerge the raw chicken breasts into the liquid. Bring to a boil, then reduce heat, cover, and simmer for 20 minutes.
Shred the chicken. Remove the cooked chicken breasts and shred them using two forks. Return the shredded meat to the pot.
Finish and serve. Stir in the fresh lime juice and simmer uncovered for 5 more minutes. Serve hot with desired toppings.