Heat a large pot over medium-high heat and then add oil. Stir in onions and garlic and cook until translucent, about 1 minute.
Add mushrooms, carrots, celery, and potatoes and cook for about 3 minutes. Add the cooked turkey pieces, salt, pepper, thyme, and oregano. Stir and cook until mushrooms become soft.
Whisk cornstarch into the cream or milk. Then add the broth and the cream/milk mixture into the soup. Stir and bring to soup to boil to allow the soup to thicken, then immediately lower heat to a simmer.
Simmer soup for 10-15 minutes, or until the vegetables are cooked, soup is thicker and the flavors are combined. Stir in minced parsley and add additional salt to taste.