Cook the Mushrooms: Add the sliced mushrooms to the pot and cook for 5-7 minutes, stirring occasionally, until they release their moisture and start to brown.
Add the Rice and Broth: Stir in the wild rice, thyme, rosemary, bay leaf, and vegetable broth. Bring the soup to a boil, then reduce the heat to low, cover, and simmer for 45-50 minutes, or until the rice is tender.
Add Creaminess: Once the rice is cooked, stir in the non-dairy milk and nutritional yeast (if using). Let the soup simmer for another 5-10 minutes. Adjust the seasoning with salt and pepper.
Serve: Remove the bay leaf and garnish with fresh parsley before serving. Enjoy with crusty bread or a side salad!