In a blender or food processor, combine the cashews, cooked potato (feel free to leave the skin on), and white beans until completely smooth, stopping to scrape down the sides as needed. This may take a few minutes. Add some of the water or broth as needed to blend.
In a large saucepan, heat the oil over medium heat. Add the onion, garlic, and celery, along with a sprinkle of salt, and stir. Continue cooking until softened, about 3 minutes.
Add the wild rice and brown rice and stir. Cook for another 1-2 minutes.
Add the rest of the broth/water, and bring the mixture to a boil. Reduce to a fast simmer and cook for about 15 minutes, or until wild rice is significantly softened.
Whisk the miso together with a bit of water to thin it out. Add it to the pot along with the vinegar, nutritional yeast, white wine if using, and black pepper.
Stir in the cashew, white bean and potato mixture. If you're not using a high-speed blender, you should pour the mixture into a mesh sieve first, using a spatula to push it through.
Continue to cook the soup at a steady simmer, stirring frequently to prevent the rice from sticking to the bottom of the pan. Add more water or broth if needed, or to thin. Take the soup off the heat when the wild rice is done, approximately 15-30 minutes. This will depend on your preference for chewier vs. softer wild rice.
Season the soup to taste. This soup needs to be salted pretty aggressively to cut through the richness, but some vegetable broths are saltier than others, so just play it by ear.