Melt 2 tablespoons of butter in a large heavy-bottomed pot over medium-high heat. Add the sliced mushrooms in a single layer and cook without stirring for 3 to 4 minutes to develop a golden sear. Continue cooking, stirring occasionally, until all liquid has evaporated and mushrooms are deeply browned, about 10 to 15 minutes total. Remove and set aside.

In the same pot, melt the remaining 1 tablespoon of butter over medium heat. Add the diced onion, carrots, and celery. Cook, stirring occasionally, until softened and the onion is translucent, about 8 to 10 minutes.

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

Pour in the chicken broth and add the wild rice, cooked chicken, and reserved mushrooms. Stir to combine and bring to a boil. Reduce heat to low, cover the pot, and simmer for 20 to 30 minutes, or until the wild rice is tender and just beginning to split open. Check your rice package for timing guidance.

Remove the lid and stir in the milk or cream and freshly grated Parmesan. Cook over low heat, stirring gently, until the cheese is fully melted and the soup is creamy, about 3 to 5 minutes. Do not let the soup boil after adding dairy. Season with salt and pepper to taste.

Ladle into bowls and serve hot. Garnish with extra grated Parmesan and fresh thyme if desired.