Heat olive oil in a large pot over medium-high heat. Add andouille sausage and cook until browned, about 5-7 minutes. Remove with slotted spoon and set aside, leaving rendered fat in pot.
Add onion, celery, and green bell pepper to the pot. Cook, stirring occasionally, until softened, about 5-6 minutes. Add garlic and cook 1 minute until fragrant.
Sprinkle flour over vegetables and stir constantly for 1 minute to cook out raw flour taste.
Gradually whisk in chicken broth until smooth. Bring to a simmer.
Add diced potatoes, Cajun seasoning, thyme, and bay leaf. Reduce heat to medium-low, cover, and simmer 15-20 minutes until potatoes are tender.
Remove bay leaf. Using immersion blender, blend about half the soup in pot, leaving chunky potato pieces for texture. Alternatively, transfer 2 cups soup to blender, blend smooth, and return to pot.
Stir in cooked sausage and heavy cream. Taste and season with salt and pepper as needed, starting with less salt since andouille is quite salty. Heat through 2-3 minutes without boiling.
Ladle soup into bowls and garnish with fresh parsley before serving.
Notes
Browning the sausage first adds smoky depth to the entire soup. Don’t skip this step.
For spicier soup, use hot andouille and add cayenne pepper or diced jalapeños with vegetables.
Substitute half-and-half for heavy cream for lighter version, though soup will be less rich.
If no immersion blender, mash some potatoes against pot side with spoon for creamier texture.
Soup thickens when refrigerated. Add splash of broth or cream when reheating.