Heat the olive oil in a Dutch oven over medium heat. Sauté the diced onion and poblano peppers for 6 to 8 minutes until soft.
Add the minced garlic, cumin, smoked paprika, and oregano. Cook while stirring for 1 minute to toast the spices.
Pour in the chicken broth, rinsed black beans, and fire-roasted tomatoes. Stir well.
Submerge the whole chicken breasts into the liquid, add the salt, and bring to a gentle boil. Reduce heat, cover, and simmer for 20 to 25 minutes.
Remove the cooked chicken, shred it with two forks on a cutting board, and return the meat to the pot.
Remove from heat and stir in the fresh cilantro and lime juice before serving.