In a large Dutch oven or heavy-bottomed pot, melt the butter over medium-high heat. Add the onion, carrots, and celery. Sauté for 5-6 minutes until the vegetables begin to soften.
Add the sliced mushrooms and cook for another 4-5 minutes until browned. Stir in the minced garlic, rosemary, and thyme, cooking for 1 minute until fragrant.
Sprinkle the flour over the vegetables and stir constantly for 2 minutes to cook out the raw flour taste (this creates the roux).
Gradually whisk in the chicken broth, ensuring no lumps of flour remain. Stir in the rinsed wild rice and the chicken pieces.
Bring the soup to a boil, then reduce the heat to low. Cover and simmer for 40-50 minutes, or until the rice is tender and the chicken is cooked through.
Stir in the heavy cream and simmer uncovered for an additional 5 minutes to thicken slightly. Season generously with salt and black pepper.
Garnish with fresh parsley and serve hot with sides of toasted bread.