Heat 1 tablespoon oil in a large Dutch oven over medium heat. Add the sliced sausage and cook for 5–7 minutes until browned and slightly crisp. Remove and set aside. Add the chicken pieces to the same pot with 1 teaspoon Cajun seasoning and cook until golden and cooked through. Transfer to the bowl with the sausage.

Melt the butter in the pot. Add the onion, celery, bell pepper, and jalapeño. Cook for about 5 minutes, stirring occasionally, until softened. Stir in the garlic, remaining Cajun seasoning, and cumin. Cook 1–2 minutes until fragrant.

Pour in the chicken broth, beans, and salsa verde. Return the sausage and chicken to the pot. Stir well and bring to a gentle simmer. Cook uncovered for 15–20 minutes so the flavors meld.

Turn off the heat. Stir in the sour cream and Parmesan until smooth and creamy. Adjust seasoning with salt and pepper to taste.

Ladle into bowls. Add a squeeze of lime juice, a sprinkle of cilantro or parsley, and sliced green onions for a fresh finish.