In a large pot or Dutch oven, melt the butter over medium heat. Add the diced onion, green bell pepper, and celery. Cook until the vegetables are softened, about 5-7 minutes. Stir in the minced garlic and cook for an additional 1 minute until fragrant.

Add the sliced andouille or smoked sausage to the pot. Cook until the sausage is browned and releases its flavors, about 5 minutes.

Sprinkle the all-purpose flour over the mixture. Stir continuously for about 2 minutes to eliminate the raw flour taste. Add the Cajun seasoning and mix well to combine.

Stir in the diced potatoes. Pour in the chicken broth, scraping the bottom of the pot to release any browned bits. Bring the soup to a boil, then reduce the heat to low. Cover and simmer for 20-25 minutes, or until the potatoes are tender.

For a creamier texture, use an immersion blender to puree a portion of the soup directly in the pot. Alternatively, remove 2 cups of the soup, blend until smooth, and return it to the pot. Be cautious when blending hot liquids.

Stir in the heavy cream or half-and-half. Add the Worcestershire sauce. Taste and adjust the seasoning with additional Cajun seasoning or salt if needed. Simmer uncovered for an additional 5 minutes to let the flavors meld.

Ladle the hot soup into bowls. Garnish with chopped green onions, shredded cheddar cheese, crumbled bacon, and fresh parsley if desired.