Begin by chopping the onion, peeling and chopping the carrots, and chopping the celery. I like to keep the carrot and celery pieces slightly rough for extra texture in the soup.
In a large soup pot, heat the olive oil and butter over medium-high heat. Add the chopped onion, carrots, and celery to the pot. Sauté the vegetables for about 7 minutes, stirring occasionally, until they begin to soften and become fragrant.
Add the minced garlic, Italian seasoning, and flour to the pot with the sautéed vegetables. Stir well and cook for 1-2 minutes until the mixture is fragrant and the flour is well incorporated. Slowly whisk in the chicken broth, ensuring the flour dissolves completely with no lumps. Stir in the uncooked wild rice.
Turn the heat to high and bring the soup mixture just to a boil. Once boiling, reduce the heat to maintain a gentle boil with the lid slightly ajar. Simmer for 15 minutes, stirring occasionally. During this time, shred or dice the cooked turkey and slice the cremini mushrooms if not already done.
Add the prepared turkey and mushrooms to the soup. Bring the soup just to a boil again, then reduce the heat to simmer with the lid slightly ajar. Continue to cook for another 15-20 minutes, or until the wild rice is tender. I like to taste a bit of rice at this stage to check for doneness—it should still have a slight chew.
Remove the pot from heat and stir in the heavy cream if using. Season the soup generously with salt and black pepper to taste. For a richer flavor, I sometimes add an extra pinch of Italian seasoning at the end.