Preheat a large soup pot to medium-high with a drizzle of olive oil. Add sausage and cook until crumbly and brown, breaking it up with a spoon as you go.
Add carrot, celery, onion, and garlic to the sausage. Stir to combine and cook veggies for about 5 minutes. Add oregano, rosemary, salt, wild rice, stock, and water to the pot. Stir everything together and cover. Reduce heat to low and let soup simmer for about 1 hour, checking it every 20 minutes, until rice is cooked through and tender. Each time you stir the soup, taste the broth and add a few pinches of salt as needed.
When rice is tender, whisk cream with cornstarch to create a slurry. Pour mixture into the soup and increase heat to medium-high. Let soup bubble uncovered for a few minutes to thicken slightly.
Serve with something green on top, if you’d like. Enjoy!