In a Dutch oven or large pot heat the oil over medium-high heat add the carrots, onions, celery, and garlic and cook, stirring often for 4-5 minutes allowing the vegetables to soften, but not brown. If they begin to brown turn the heat down.
Add the wild rice, mushrooms, bay leaf, thyme, kale, vegetable broth, salt, and pepper and increase the temperature to high and bring to a boil. Reduce the heat to medium and cook for about 45 minutes or until the the rice is thoroughly cooked.
Add the coconut milk and simmer on low for 10 more minutes.