Melt 2 tbsp butter in a pan over medium-high heat.
Add mushrooms and cook until they release their liquids and the liquid evaporates, about 10-15 minutes. Set aside.
Melt 1 tbsp butter in the same pan. Add onion, carrots, and celery. Cook until tender, about 8-10 minutes.
Mix in garlic and thyme. Cook until fragrant, about 1 minute.
Add chicken broth, wild rice, chicken, and mushrooms. Bring to a boil. Reduce heat, cover, and simmer until rice is tender, about 20-30 minutes.
Stir in milk and Parmesan cheese. Cook until cheese melts, and season with salt and pepper to taste.