Heat oil in a large Dutch oven over medium heat. Add onion and cook until soft and translucent, about 5 minutes, then add garlic and stir for 1 minute.

Add cumin, oregano, and salt to the pot and stir for 30 seconds. Scrape in the green chiles with their liquid and cook until fragrant.

Pour in the chicken broth and add the drained beans. Nestle in the chicken breasts, bring to a boil, then reduce heat, cover, and simmer until chicken is cooked through, 18 to 20 minutes.

Remove the chicken, shred it with two forks, and return it to the pot. Whisk masa harina with ¼ cup cold water to make a slurry, then stir it into the chili along with the cubed cream cheese. Simmer over low heat until thickened, about 5 minutes. Taste and adjust salt.

Ladle into bowls and top with tortilla chips, avocado, cilantro, and a squeeze of lime.