Heat oil over medium heat in a large soup pot or dutch oven. Cook onion, carrots, mushrooms, celery, and garlic until starting to soften, about 5 minutes.
Add broth, rice, parsley, thyme, black pepper, sage, bay leaf, and salt. Cover, bring soup to a simmer, and continue cooking at a simmer until the rice is done.
In a medium bowl, whisk cornstarch into milk. Add to soup, along with the turkey, and bring soup up to a boil for about 5 minutes to thicken and heat through.