Step 1: Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the sliced Andouille sausage and cook until browned and crispy, about 5-7 minutes. Remove the sausage with a slotted spoon and set aside, leaving the rendered fat in the pot.
Step 2: Reduce heat to medium. Add the diced onion, celery, and green bell pepper to the pot. Sauté for 5-7 minutes until softened, scraping up any browned bits from the bottom of the pot.
Step 3: Add the minced garlic, dried thyme, dried oregano, Cajun seasoning, black pepper, and cayenne pepper (if using). Cook for 1 minute more until fragrant.
Step 4: Pour in the chicken broth and add the diced potatoes. Bring the mixture to a boil, then reduce heat to a simmer, cover, and cook for 15-20 minutes, or until the potatoes are fork-tender.
Step 5: Remove the pot from the heat. Using an immersion blender, carefully blend about two-thirds of the soup until creamy. For a chunkier soup, you can mash some of the potatoes with a potato masher instead. If you don't have an immersion blender, transfer portions of the soup to a regular blender (being careful with hot liquids) and blend until smooth, then return to the pot.
Step 6: Stir in the heavy cream and milk. Add the reserved cooked Andouille sausage back into the soup. Taste and season with salt as needed.
Step 7: Gently heat the soup over low heat until warmed through, but do not boil after adding the cream. Serve hot, garnished with fresh chopped parsley and sliced green onions.