Sauté Vegetables: In a large pot, heat the olive oil over medium heat. Add the onion, carrots, and celery, and sauté until softened, about 5 minutes. Add the garlic and mushrooms, and cook for an additional 3-4 minutes until the mushrooms are tender.
Cook Chicken: Add the chicken pieces to the pot and cook until browned on all sides, about 5-7 minutes.
Add Rice and Broth: Stir in the wild rice, dried thyme, and dried rosemary. Pour in the chicken broth and bring to a boil. Reduce heat to low, cover, and simmer for about 45 minutes, or until the rice is tender.
Make Creamy Base: In a small bowl, whisk together the heavy cream and flour until smooth. Slowly stir this mixture into the soup. Continue to simmer for another 10 minutes, stirring occasionally, until the soup has thickened.
Season and Serve: Season the soup with salt and pepper to taste. Serve hot, garnished with fresh parsley if desired.