Place a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the sliced andouille sausage and cook, stirring occasionally, until browned and crispy, about 5-7 minutes. Use a slotted spoon to remove the sausage to a paper towel-lined plate, leaving the rendered fat in the pot.

Reduce heat to medium. Add the butter to the pot. Once melted, add the diced onion, celery, and bell pepper (the 'holy trinity'). Sauté for 6-8 minutes, until the vegetables have softened. Add the minced garlic and cook for another minute until fragrant.

Sprinkle the all-purpose flour, Cajun seasoning, smoked paprika, and dried thyme over the vegetables. Stir constantly for 1-2 minutes to cook out the raw flour taste and create a roux.

Slowly pour in 1 cup of the chicken broth, scraping the bottom of the pot to deglaze and incorporate any browned bits. Once smooth, gradually whisk in the remaining 5 cups of broth.

Add the cubed potatoes and the bay leaf to the pot. Increase the heat to bring the soup to a low boil, then reduce the heat to maintain a steady simmer. Cover and cook for 15-20 minutes, or until the potatoes are very tender and easily pierced with a fork.

Remove the pot from the heat and discard the bay leaf. Use an immersion blender to blend the soup to your desired consistency. For a creamier soup, blend until smooth. For a chunkier texture, pulse only a few times, leaving plenty of potato pieces intact.

Return the pot to low heat. Add the softened cream cheese and heavy cream. Stir gently until the cream cheese is fully melted and the soup is smooth and creamy. Do not let it boil.

Stir the cooked andouille sausage back into the soup and heat through for a few minutes. Taste and season with additional salt and black pepper if needed. The sausage and Cajun seasoning are often salty, so be sure to taste before adding more.

Ladle the hot soup into bowls and serve immediately, topped with your favorite garnishes like sliced green onions and shredded cheddar cheese.