Warm the butter and olive oil in your pot over medium heat, then add the diced onion and jalapeño. Let them soften for about 4 minutes, then stir in minced garlic and cook another minute until fragrant.
Season the chicken lightly with salt and pepper, then place it in the pot. Pour in enchilada sauce, diced tomatoes (with juices), black beans, corn, hot sauce (if using), chicken broth, and the seasoning mix you prepared. Bring to a gentle boil, then reduce to a simmer.
After simmering about 15–20 minutes (or until the chicken is cooked through), remove the chicken and shred it using two forks. Return it to the soup. Lower heat, stir in softened cream cheese until smooth, then melt in the shredded cheddar and Monterey Jack until fully incorporated. Adjust seasoning and serve.