Heat a large pot over medium heat and add a drizzle of oil. Once the oil shimmers, add sliced smoked sausage and cook until golden brown and slightly crispy, about 5–7 minutes. Stir occasionally to ensure even browning. Once ready, remove the sausage from the pot and set it aside, leaving the delicious drippings.

In the same pot, season chicken breasts or thighs generously with Cajun seasoning. Cook over medium heat for 5–7 minutes, turning occasionally, until the chicken is golden brown on all sides. Once seared, transfer the chicken to a plate and set aside.

Melt 1 tablespoon of butter in the pot, scraping up any flavorful bits left from the sausage and chicken. Add diced onion, celery, bell pepper, and jalapeño. Sauté for 7–10 minutes until the vegetables are tender and fragrant.

Stir in minced garlic, ground cumin, and a pinch of Cajun seasoning. Cook for about 1 minute until the garlic becomes fragrant.

Pour in 4 cups of chicken broth, followed by the white beans and salsa verde. Bring the mixture to a soft boil, then reduce the heat to low, allowing it to simmer gently for 15 minutes.

After simmering, stir in ½ cup of sour cream and ¼ cup of grated Parmesan cheese. Mix well until everything is creamy and blended, about 2–3 minutes.

Remove the pot from heat and squeeze in fresh lime juice for a bright finish. Stir in chopped cilantro or parsley, and sprinkle with sliced green onions if desired. Taste the chili and season with salt and pepper as needed.