Sauté the aromatics: Heat olive oil in a large pot over medium heat and cook the onion until soft, about 4 minutes. Add the garlic, green chilies, cumin, oregano, chili powder, and cayenne, and stir for 1 minute until fragrant.
Add broth and chicken: Pour in the chicken broth and add the whole chicken breasts, submerging them as much as possible. Bring to a gentle simmer over medium heat.
Poach until cooked: Simmer the chicken for about 15 minutes, until it reaches an internal temperature of 165°F. Remove the chicken to a plate and let it rest for a few minutes.
Shred the chicken: Use two forks to shred the chicken into bite-sized pieces. The warm chicken pulls apart easily at this stage.
Return chicken to pot: Add the shredded chicken back into the pot along with the beans and corn. Stir everything together so it's evenly distributed.
Simmer to thicken: Let the chili simmer uncovered over low heat for 10 minutes, stirring occasionally. The broth should reduce slightly and the beans will start to soften.
Mash some beans: Press the back of a spoon against a few beans along the side of the pot to mash them. This releases starch and naturally thickens the chili.
Turn off the heat: Remove the pot from direct heat before adding the dairy. This keeps the yogurt from curdling.
Stir in yogurt and cheese: Add the Greek yogurt and shredded Monterey Jack cheese, stirring until the cheese melts and the chili turns creamy and pale.
Season and rest: Taste and adjust with salt and black pepper. Let the chili rest for 10 minutes off heat; it thickens further as it sits.
Serve warm: Ladle the chili into bowls and top with fresh cilantro and a squeeze of lime juice. Serve immediately while warm.