Sauté the aromatics: In a large dutch oven, add the diced onion, celery, and carrots. Cook for 5–7 minutes, stirring occasionally, until the vegetables are softened. Add the minced garlic and cook for 1 minute, until fragrant.
Add the potatoes and broth: Stir in the diced potatoes, thyme, and a generous pinch of salt and pepper. Pour in the chicken broth, ensuring the potatoes are fully submerged. Bring the mixture to a boil, then reduce the heat to medium-low and let it simmer for 20 minutes, or until the potatoes are fork-tender.
Stir in the cream and sausage: Add the heavy cream to the pot, stirring to combine. Add the cooked sausage to the soup and simmer for an additional 10 minutes to allow the flavors to meld. Adjust seasoning with salt and pepper, as needed.
Serve and garnish: Ladle the soup into bowls and garnish with freshly chopped green onion, if desired. Serve warm with crusty bread or crackers on the side.