Drain and rinse the beans. Mash one can in a bowl with a potato masher or fork, adding a few tablespoons of the measured broth to loosen it. Leave the other two cans whole.

Heat olive oil in a 5- to 6-quart pot over medium heat. Cook onion for 5–6 minutes until softened. Stir in garlic, cumin, and oregano for 30 seconds.

Add remaining broth, green chiles, whole chicken breasts, salt, and pepper. Bring to a gentle simmer, partly cover, and cook 18–25 minutes, until each chicken breast reaches 165°F in the thickest part. Remove cooked chicken to a clean plate.

Stir mashed beans, whole beans, and frozen corn into the pot. Simmer gently for 8–10 minutes, stirring across the bottom to prevent sticking. Shred cooked chicken with two forks while the beans heat.

Return shredded chicken to the pot and stir until evenly hot. Remove from heat and stir in lime juice and most of the cilantro. Add a little hot water if too thick. Divide into six bowls and finish with remaining cilantro and lime wedges.

Refrigerate leftovers in shallow containers within 2 hours, or 1 hour above 90°F. Use within 3–4 days. Reheat with a little water as needed, stirring until the entire portion reaches 165°F.