Prepare the vegetables by dicing the onion, bell pepper, carrots, and celery, and mincing the garlic.
In a large pot, heat the olive oil over medium heat. Add the diced onion, garlic, bell pepper, carrots, and celery. Sauté for 5-7 minutes until softened.
Add the diced potatoes, chicken broth, Cajun seasoning, dried thyme, salt, and pepper. Bring to a boil.
Reduce heat to low, cover, and let simmer for 20-25 minutes until potatoes are tender. Mash some potatoes for a thicker consistency.
Stir in the heavy cream and sausage (if using), and heat through for an additional 5 minutes. Adjust seasoning if needed, then serve garnished with green onions and parsley.