In a large pot, melt 2 tablespoons of butter over medium-high heat. Add the sliced mushrooms and cook until they release their moisture and the liquid evaporates, about 10-15 minutes. Set the mushrooms aside.

In the same pot, melt the remaining 1 tablespoon of butter. Add diced onion, carrots, and celery. Sauté until the vegetables are softened, around 8-10 minutes.

Stir in the minced garlic and chopped thyme. Cook until fragrant, about 1 minute.

Pour in the chicken broth, then add the wild rice, cooked chicken, and sautéed mushrooms. Bring the soup to a boil, then reduce the heat and let it simmer, covered, for 20-30 minutes, or until the rice is tender.

Stir in the milk or cream and grated parmesan cheese. Continue to simmer until the cheese is fully melted and the soup is creamy.

Taste the soup and season with salt and pepper as needed. Serve hot.