Melt the butter in a large pot or Dutch oven over medium heat. Then add the onion, carrots, and celery. Cook them for 6 to 7 minutes, until they begin to soften and smell sweet.

Add the sliced mushrooms and cook for 5 to 6 minutes. Let them release their liquid, and keep cooking until that liquid evaporates and the mushrooms turn golden. Then stir in the garlic and cook for 1 minute, just until fragrant.

Sprinkle the flour over the vegetables and stir well. Cook for 1 to 2 minutes, stirring constantly, so the raw flour taste cooks out. After that, slowly pour in the broth while whisking to help prevent lumps.

Add the rinsed wild rice, bring the soup to a boil, and then reduce the heat to low. Simmer it partially covered for 35 to 40 minutes, until the wild rice is tender and many of the grains have split open.

Stir in the cooked turkey and cook for 3 to 4 minutes, until it is heated through. Then lower the heat and add the heavy cream, thyme, salt, and pepper. Simmer gently for 5 minutes, but do not let it boil hard.

Taste the soup and adjust the seasoning as needed. Then ladle it into bowls and finish with fresh parsley on top. Serve it hot.