Heat the butter and olive oil in a large Dutch oven over medium to medium high heat. Add the sliced mushrooms and cook for 10 to 14 minutes, stirring occasionally, until they release their moisture and begin to brown deeply.
Add the onion, carrots, and celery to the pot. Cook for 6 to 8 minutes until softened. Stir in the garlic and thyme and cook for 30 seconds until fragrant.
Sprinkle the flour over the vegetables and mushrooms. Stir well to coat everything evenly, then cook for 1 minute to remove the raw flour taste.
Slowly pour in the chicken broth while stirring. Add the rinsed wild rice and bring the soup to a gentle boil. Reduce the heat to a simmer, cover partially, and cook for 35 to 45 minutes, or until the wild rice is tender but still pleasantly chewy.
Stir in the cooked chicken during the last 10 minutes of cooking so it warms through without drying out.
Lower the heat so the soup is no longer boiling. Stir in the half and half, heavy cream, and Parmesan cheese until the broth is smooth and creamy.
Season with kosher salt and black pepper to taste. If the soup is too thick, add a splash of broth. If it is too thin, simmer uncovered for a few more minutes.
Let the soup rest for 3 to 5 minutes before serving. Ladle into bowls and top with parsley, extra Parmesan, black pepper, or fresh thyme if desired.