1 cup raw cashews
1 bunch carrots (about 10 oz.), trimmed, peeled, cut into 1" pieces
1 large fennel bulb (about 10 oz.), fronds reserved, bulbs halved, cored, and cut into 1" pieces
4 Tbsp. extra-virgin olive oil, divided
1¼ tsp. kosher salt, divided, plus more
Freshly ground white or black pepper
1 medium onion, finely chopped
2 tsp. finely chopped peeled ginger
1 Tbsp. mirin
2½ cups (or more) no- or low-sodium vegetable broth
2 tsp. plus 1 Tbsp. white miso
4 oz. shiitake mushrooms (about 4 large), thinly sliced
1 tsp. unseasoned rice vinegar
Place cashews in a small bowl and pour in water to cover. Let soak at room temperature at least 2 hours and up to 4 hours.
Preheat oven to 400°. Toss carrots and fennel on a rimmed baking sheet with 2 Tbsp. oil and 1 tsp. salt. Season with pepper. Roast, rotating pan halfway through, until vegetables are just starting to brown around the edges, about 20 minutes.
Heat 1 Tbsp. oil in a large Dutch oven or other heavy pot over medium-high. Add onion, ginger, and ¼ tsp. salt and cook, stirring occasionally, until onion is softened slightly and translucent, about 5 minutes. Mix in roasted carrots and fennel, then add mirin, scraping up any browned bits. Pour in broth and bring to a boil. Reduce heat, cover, and simmer until vegetables are softened, 30–40 minutes.
Meanwhile, whisk 2 tsp. miso and 1 Tbsp. warm water in a small bowl to combine. Heat remaining 1 Tbsp. oil in a medium skillet over high. Cook mushrooms, tossing occasionally, until golden brown, about 6 minutes. Remove from heat, add miso mixture, and stir to coat. Set aside.
Drain cashews and transfer to a blender. Pour in enough water to cover (½–¾ cup) and blend on high speed until smooth and light. (about 2 minutes if you’re using a high-speed blender). It should be the consistency of just-whipped cream. Thin with more water if needed.
Remove pot with carrot mixture from heat and stir in remaining 1 Tbsp. miso. Purée soup with an immersion blender until smooth (or, use a regular blender, working in batches if needed). Thin with more broth or water, if needed. Stir in vinegar; taste and season with more salt and pepper if needed. Stir in ¼ cup cashew cream.
Ladle soup into bowls and top with reserved mushrooms, a dollop of cashew cream, and reserved fennel fronds.