Roast the poblanos until blistered, steam them in a covered bowl, then peel, seed, and dice the flesh.
Sauté the diced onion in olive oil until soft, then stir in the garlic, cumin, oregano, and smoked paprika for 1 minute.
Combine the diced poblanos, chicken broth, rinsed black beans, and tomatoes in the pot, then submerge the raw chicken breasts.
Simmer covered for 20 to 25 minutes until the chicken is tender enough to shred with two forks.
Stir the shredded meat back into the soup along with the fresh lime juice, simmer for 5 minutes, and serve garnished with cilantro.