Heat the olive oil in a large pot or Dutch oven over medium heat. Add the onion and cook about 5 minutes until soft, then stir in the garlic, cumin, and oregano and cook 1 minute more.
Pour in the broth, green chiles, and corn. Nestle in the chicken breasts, bring to a gentle simmer, and cook 12-15 minutes, until the chicken reaches 165°F in the center.
Move the chicken to a cutting board and shred with two forks; set aside. (If using rotisserie chicken, skip this and stir it in at the end of step 5.)
Add the beans to the pot and mash about 1 cup of them against the side of the pot to thicken the broth. Simmer uncovered 10 minutes.
Return the shredded chicken to the pot and warm through, 1-2 minutes, then remove the pot from the heat.
Stir in the Greek yogurt and cilantro off the heat so the yogurt stays smooth. Serve topped with extra yogurt and cilantro.