01 - Heat olive oil and butter in a large pot over medium heat. Add sliced Andouille sausage and cook until browned, about 4–5 minutes. Remove sausage and set aside.
02 - In the same pot, add onion, celery, carrot, bell pepper, and garlic. Sauté for 5–6 minutes until vegetables are soft.
03 - Stir in Cajun seasoning, smoked paprika, thyme, cayenne, and a pinch of salt and black pepper. Cook for 1 minute to toast spices.
04 - Add diced potatoes and chicken broth. Bring to a boil, then reduce heat to a simmer. Cover and cook for 15–20 minutes, until potatoes are very tender.
05 - Using a potato masher or immersion blender, partially mash some of the potatoes in the pot to thicken the soup, leaving some chunks for texture.
06 - Return sausage to the pot. Stir in the milk and simmer for 5–10 minutes longer.
07 - For extra thickness, mix cornstarch with cold water to make a slurry. Stir into the soup and simmer for 2–3 minutes until thickened.
08 - Taste and adjust seasoning with more Cajun seasoning, salt, or pepper if needed. Ladle soup into bowls and garnish with sliced green onions.