In a large pot, heat the olive oil over medium heat. Add the chopped onion, celery, and carrots. Sauté for about 5 minutes until the vegetables are tender.
Add the minced garlic and cook for an additional minute until fragrant.
Stir in the diced potatoes and Cajun seasoning, mixing well to coat the potatoes.
Pour in the chicken broth and bring the mixture to a boil. Reduce heat and let it simmer for 20-25 minutes, or until the potatoes are tender.
Using a potato masher or immersion blender, mash or blend the soup to your desired consistency. If you prefer a chunkier soup, mash only half of the potatoes.
Stir in the heavy cream and season with salt and pepper to taste. Heat through for another 5 minutes.
Serve hot, garnished with sliced green onions and shredded cheddar cheese if desired.