To begin, heat the butter and oil in your Dutch oven or soup pot over medium heat. Toss in the chicken, onion jalapeño pepper, and garlic until the chicken is opaque and the onions are tender. About 5 minutes.
Add in the oregano, cumin, cayenne pepper, salt, black pepper, green chilies, and broth.
Bring the soup to a boil, reduce the heat, and cook for about 10 - 15 minutes, or until the chicken is cooked through.
Turn the heat up for a few minutes to bring the soup to a boil.
Whisk the cold water and corn starch together in a small dish. Add half of the mixture to the boiling soup to thicken. If you would like it thicker, add the rest of the cornstarch slurry.