In a large dutch oven, heat olive oil over medium-high heat. Add onion and stir, cooking for two minutes. Then add celery, carrots, mushrooms, and salt. Stir to coat, and cook for 8 minutes to soften all the vegetables. Add oregano, black pepper, and wild rice, stirring to combine.
Add vegetable broth and thyme. Bring to a boil, then reduce heat to a simmer and cover. Cook for 45 minutes, until wild rice is popped and tender to the bite.
In a small bowl, whisk together cornstarch and half and half, so there are no lumps. Then whisk in Dijon mustard. Turn heat down on soup so it is not simmering, and slowly add liquid mixture. Stir to combine, then add tofu and gently stir.
Let cook on low for 10 minutes so it thickens and tofu absorbs flavors. Stir in parsley and serve with crackers or bread.