Heat olive oil in a Dutch oven over medium heat. Cook chicken, onion, and garlic in hot oil until chicken is browned on both sides, about 5 minutes.

Transfer chicken to a cutting board; cut into 1-inch pieces.

Return chicken pieces to the Dutch oven. Add chicken broth, cannellini beans, green chiles, oregano, cumin, and cayenne pepper. Bring mixture to a simmer and cook until chicken is cooked through, 30 to 40 minutes.

Divide cilantro among 4 bowls. Ladle chili over cilantro and top with cheese.

Season with salt to serve.