Prepare the onion, carrots, and celery by chopping them. The carrots and celery can be roughly chopped.
In a large soup pot, heat the olive oil and butter over medium-high heat. Add the onion, carrots, and celery, and sauté for about 7 minutes, stirring occasionally.
Add the garlic, Italian seasoning, and flour to the pot. Cook while stirring for 1 to 2 minutes.
Gradually whisk in the chicken broth, mixing until the flour is fully dissolved.
Stir in the wild rice.
Turn the heat up to high and bring the soup to a boil. Once it reaches a boil, cover the pot with the lid slightly ajar, reduce the heat to maintain a gentle boil, and cook for 15 minutes.
If you have not already prepared the turkey and mushrooms, do so while the rice is cooking.
Add the turkey and mushrooms to the soup. Increase the heat just until the soup boils again, then reduce it to a simmer. Cover the pot with the lid slightly ajar, and continue to cook for another 15 to 20 minutes, or until the rice is tender (wild rice will retain a bit of chewiness).
If desired, stir in the heavy cream. Season the soup with salt and pepper to taste before serving.