In a large pot, warm 2 tablespoons oil over medium heat. Add ginger, garlic, and white parts of scallions. Cook for a minute, stirring often, until fragrant but not browned. Stir in 2 tablespoons miso paste, tomato paste, and gochujang/chili paste and cook for 3-4 minutes, stirring often, or until tomato paste is caramelized and sticking to the bottom of the pot.
Stir 4 cups of water into the pot, along with the dried mushrooms (if using). Scrape up anything stuck to the bottom of the pot. Bring to a boil over high heat, then reduce the heat to low and simmer for 10 minutes. Remove the mushrooms (if using, set aside to cool, then squeeze out any liquid back into the pot, then slice up to top the soup, leaving out the woody stems).
Meanwhile, place soaked cashews and remaining 4 cups of water into a high-speed blender and blend on high until smooth, about 1 minute. Pour into the pot and bring back to a simmer.
Add remaining ¼ cup miso into a ladle or large measuring up, add some of the hot soup broth, and mix/mash the miso to dissolve in the liquid. Pour back into the pot and turn off the heat. Stir in the soy sauce and sugar.