Add the olive oil and butter to a Dutch oven or soup pot. Add the onion, celery and carrot and cook, stirring occasionally for 5 minutes until lightly browned. Add mushrooms and saute for 2 minutes. Add the garlic and stir for 2 minutes.
Add the vegetable broth, wild rice, spices, kosher salt, and black pepper. Bring to a simmer. Cover the pot and simmer for 45 minutes to 1 hour, until rice breaks open, stirring occasionally.
When the wild rice is cooked, stir in the milk and Parmesan cheese. Remove 2 cups from the soup using a liquid measuring cup (including veggies, rice and broth) and place it in a blender. Add the cornstarch, then blend. Pour it back into the soup.
Stir in the soy sauce, taste and adjust seasonings as desired. Enjoy warm. Stores refrigerated for 3 days.