In a large pot, heat olive oil over medium heat. Add the onion and garlic, and sauté until they are translucent.
Add the sliced mushrooms and cook until they are tender and have released their moisture.
Stir in the wild rice, ensuring it is well coated with the oil and mushroom mixture.
Pour in the vegetable broth and bring the soup to a boil. Reduce the heat and let it simmer, covered, for about 50 minutes, or until the rice is tender.
Stir in the heavy cream and season with salt and pepper to taste. Add the fresh thyme and let the soup simmer for an additional 10 minutes.
Serve hot, garnish with additional thyme if desired.