Place a large pot over medium heat and add the oil. Once hot, sauté the diced andouille or smoked turkey sausage for about 5 minutes until well browned and crispy. Remove and set aside. In the same pot, add the chopped onion, celery, and green bell pepper. Cook for 5–6 minutes, stirring occasionally, until the vegetables are soft. Add the minced garlic, Cajun seasoning, smoked paprika, cayenne pepper (if using), and cornstarch. Stir well to coat all the vegetables in the spices and let them cook for another 2 minutes.

Add the chicken broth gradually, stirring constantly so the cornstarch dissolves completely and no lumps remain. Then stir in the shredded chicken, white beans, kidney beans, and the previously toasted sausage. Lower the heat, cover, and simmer for 15–20 minutes to allow the flavors to meld. Season with salt and pepper to taste.

Turn off the heat and add the lime juice and Greek yogurt. Stir well to integrate, being careful not to boil once the yogurt is added so it doesn't curdle. Let sit for a few minutes, then serve hot, garnished with chopped cilantro, lime wedges, and a touch of shredded cheese if desired.