Sauté the vegetables. In the same pot, add the butter, diced onion, carrots, and celery. Cook for about 5 minutes, stirring often, until the onion softens and the vegetables begin to turn tender. Add the garlic and cook for 30 seconds more, just until fragrant.

Create the flavor base. Sprinkle in the flour, Cajun seasoning, thyme, smoked paprika, and black pepper. Stir constantly for about 1 minute so the flour cooks through and coats the vegetables. This helps give the soup that silky, creamy body later on.

Add the potatoes and broth. Pour in the chicken broth while stirring to prevent lumps. Add the cubed potatoes and return the browned sausage to the pot. Increase the heat slightly and bring the mixture to a gentle boil.

Simmer until tender. Reduce the heat to medium-low, cover loosely, and simmer for 15 to 18 minutes, or until the potatoes are fork-tender. If you want an even creamier texture, use the back of your spoon to lightly mash some of the potatoes against the side of the pot.

Make it creamy. Lower the heat so the soup is no longer boiling. Stir in the milk and heavy cream. Let the soup warm through gently for 2 to 3 minutes. Avoid boiling at this stage so the dairy stays smooth.

Add the cheese. Stir in the shredded cheddar a handful at a time, mixing well after each addition until melted and fully incorporated. Taste the soup and add salt if needed. Because andouille and broth can already be salty, I always recommend tasting first before adding extra.

Serve and garnish. Ladle the soup into bowls and top with green onions, extra cheddar, and a little black pepper if you like. Serve hot with crusty bread for dipping. It’s rich, savory, and incredibly comforting.