In a large saucepan over medium heat, heat a tablespoon of oil. Once shimmering, add one pound of sliced andouille sausage, cooking for 5-7 minutes until golden brown and fragrant. Stir occasionally with a wooden spoon, allowing the sausage to crisp up. Once cooked, remove the sausage from the pan and set it aside for later.

In the same pan, season one pound of boneless chicken breasts or thighs with Cajun seasoning. Add the chicken to the pan and cook for about 5-7 minutes on each side until golden and fully cooked. Use tongs to flip the chicken, ensuring an even browning. Once done, remove the chicken from the pan and let it rest before shredding.

In the same saucepan, melt one tablespoon of butter over medium heat. Add one cup of diced onion, one cup of diced celery, one cup of diced bell pepper, and one finely diced jalapeño. Cook for 7-10 minutes until the vegetables are tender, stirring frequently.

To the tender vegetable mixture, stir in four chopped garlic cloves, one teaspoon of ground cumin, and an additional tablespoon of Cajun seasoning. Cook for about one minute until the garlic becomes fragrant, stirring continuously to prevent burning.

Pour in four cups of chicken broth, followed by two cans of rinsed and drained white beans and half a cup of salsa verde. Return the cooked sausage and shredded chicken to the pot. Bring the mixture to a boil, then reduce the heat to low. Simmer uncovered for about 15 minutes.

Once simmering, add half a cup of sour cream and a quarter cup of grated Parmesan cheese to the pot. Stir gently until the cheese melts and integrates into the chili.

To brighten the flavors, stir in a tablespoon of lime juice, and add chopped cilantro or parsley for garnish. Season the chili with salt and pepper to taste.