Whisk together vegetable stock and arrowroot powder until dissolved, and set aside.
Heat oil, add onions and garlic, and sauté for 3 minutes. Add mushrooms, thyme, salt, and pepper and cook, stirring occasionally, for 5 minutes. Add soy sauce and vegetable broth, bring to a boil, lower heat, and simmer, stirring occasionally, for 15 minutes. Whisk in nutritional yeast until it dissolves, then whisk in soy milk for a minute. Let cool slightly before serving.