Sauté aromatics: In a heavy-bottomed pot, heat oil or butter over medium heat. Add onion, carrots, and celery. Cook 5–7 minutes until soft.
Add mushrooms and herbs: Toss in garlic, mushrooms, thyme, and rosemary. Cook for 5 more minutes until mushrooms release moisture and begin to brown.
Simmer with rice: Stir in wild rice, then pour in the vegetable broth. Bring to a boil, then reduce heat and cover. Simmer for 45 minutes or until rice is tender but chewy.
Cream it up: Stir in coconut milk or heavy cream. Simmer uncovered for 5 more minutes to let flavors marry.
Season and serve: Season with salt and pepper. Ladle into bowls, top with fresh parsley, and serve hot with crusty sourdough or garlic toast.