Heat olive oil in a Dutch oven over medium heat. Cook chicken, onion, and garlic in hot oil until the chicken is browned completely, 3 to 5 minutes per side.
Remove the chicken to a cutting board, cut into 1-inch pieces, and return to the Dutch oven; add chicken broth, cannellini, green chilies, oregano, cumin, and cayenne pepper. Bring the mixture to a simmer and cook until the chicken is cooked through, 30 to 45 minutes.
Divide cilantro among 4 bowls. Ladle chili over cilantro and top with cheese. Season with salt to serve.