Warm the butter or oil in a large dutch oven or heavy-bottomed pot over medium-high heat. Add the onion, carrot, and celery and sauté for 3 to 5 minutes, until beginning to brown. Add the mushrooms, garlic, smoked paprika, thyme, and black pepper. Sauté for an additional 2 to 3 minutes, stirring occasionally, until the mushrooms have reduced to about half their size.
Add the wild rice, vegetable broth, and 3 cups of the water to the pot. Cover and bring to a boil over high heat.
Uncover, reduce the heat to medium, and simmer for 45 to 50 minutes, stirring occasionally. Stop cooking when the majority of the wild rice has split open to look like a hot dog bun (do not overcook!). The soup should get pretty thick at this point, but if it looks too thick you can thin it out with additional water.
In the meantime, add the cashews, miso paste, and remaining 1 cup of water to a high-speed blender. Blend on high until smooth, 45 to 60 seconds.
Add the miso cream to the soup and mix well; reduce the heat to medium-low and simmer for an additional 2 to 3 minutes, until warm. Adjust the soup as necessary here (more liquid, salt, pepper, etc).
Serve warm; store any leftovers in the refrigerator for up to 5 days. The rice will absorb some of the soup’s liquid as it sits, so you may want to add extra water or vegetable broth when reheating.
