In a large pan over medium-high heat, melt butter and add the sliced mushrooms. Sauté until the mushrooms release their juices and the moisture evaporates, about 10-15 minutes. Remove and set aside.
Using the same pan, add another tablespoon of butter along with the diced onion, carrots, and celery. Cook until soft, around 8-10 minutes.
Stir in the chopped garlic and thyme, then sauté for a minute until fragrant.
Add in the chicken broth, followed by the wild rice, cooked chicken, and the sautéed mushrooms. Bring to a boil, then reduce to low and cover, letting it simmer until the rice is tender, approximately 20-30 minutes.
Finally, stir in the milk and parmesan cheese, cooking until the cheese melts. Adjust the seasoning with salt and pepper according to your preference.