1. In a large pot, melt the butter over medium heat. Add the sliced mushrooms and cook until they are softened and browned, about 5-7 minutes. Remove the mushrooms from the pot and set them aside.

2. In the same pot, melt the remaining butter. Add the diced onion, carrots, celery, and chopped garlic. Cook until the vegetables are tender, about 5-7 minutes.

3. Stir in the chopped thyme, chicken broth, and wild rice. Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for 45-50 minutes, or until the rice is tender.

4. Stir in the cooked chicken, milk or cream, and grated Parmesan. Cook for an additional 5 minutes, stirring occasionally, to allow the flavors to meld.

5. Taste the soup and adjust the seasoning with salt and pepper as needed.

6. Ladle the hot soup into bowls and serve immediately, garnished with additional thyme or Parmesan if desired.