Heat olive oil in a large pot over medium heat. Cook diced onion for 4 to 5 minutes until softened.
Stir in garlic and cook 30 seconds until fragrant.
Add chicken, broth, cumin, oregano, chili powder, salt, and pepper. Bring to a boil, then reduce heat and simmer 20 minutes until chicken is cooked through.
Remove chicken, shred with two forks, and return to the pot.
Stir in beans and green chilies. Simmer 10 minutes.
Reduce heat to low and stir in sour cream until combined.
Add cheese and stir until melted.
Adjust seasoning, then serve garnished with cilantro and lime wedges.