Heat the olive oil over medium-low heat in a large pot. Add the carrots and onion and cook, stirring occasionally, for 10 minutes. Add the sliced mushrooms and minced garlic and continue to cook, stirring occasionally, for 10 more minutes.
Add the salt, thyme, and black pepper, and cook, stirring occasionally, for 3 minutes. Add the cornstarch and nutritional yeast and mix well.
Add in the vegetable broth, ½ cup of water, rinsed wild rice, and coconut aminos, cover, and bring to a boil. Reduce heat to medium-low and cook, covered, for 1 hour or until the rice is tender, stirring occasionally.
Add the shredded kale and coconut milk, stir, and allow the kale to wilt, 1 to 2 minutes. Serve immediately or simmer over low heat until ready to serve.
