Melt butter in olive oil over medium-high heat in a large Dutch oven or soup pot. Add the sliced chicken and onions, sautéing them together for about three minutes until they start to brown slightly.
Add in the bell peppers, minced garlic, flour, and all seasonings/spices. Stir constantly for about two minutes.
Stir in the uncooked rice along with black beans, green chilies, crushed tomatoes, and three cans of chicken broth. Mix well! In a separate bowl, mix cornstarch with the last can of chicken broth before adding it to your pot. Bring everything to a boil then reduce heat to medium-low. Simmer while stirring occasionally for about eight to twelve minutes or until the rice is cooked through.
Once done cooking, remove from heat and stir in freshly grated cheese until melted. If you prefer a thinner consistency, feel free to add additional chicken broth or water at this stage. Serve warm and garnish individual bowls with your favorite toppings.