1. Cook the Sausage: In a large pot, heat the olive oil over medium heat. Add the Italian sausage and cook, breaking it into crumbles, until browned. Drain any excess grease if needed.

2. Add the Aromatics: Stir in the diced onion and minced garlic. Cook for 2-3 minutes, until the onion is softened and fragrant.

3. Add the Potatoes and Broth: Pour in the chicken broth and add the diced potatoes. Bring the mixture to a boil, then reduce the heat and let it simmer for 15-20 minutes, until the potatoes are tender.

4. Stir in the Cream: Pour in the heavy cream or half-and-half. Add the salt, pepper, and thyme (or Italian seasoning). Let the soup simmer for an additional 5 minutes to allow the flavors to meld.

5. Finish and Serve: If desired, stir in the shredded cheddar cheese. Ladle the hot Sausage Potato Soup into bowls, garnish with fresh parsley or green onions, and enjoy!