Cut sweet potatoes into smaller pieces if needed to fit in a spiralizer, then crank through to create spaghetti-size coils (save leftover chunks of sweet potato for other uses). Tear coils into manageable lengths. Heat 2 tbsp. oil in a large frying pan over medium-high heat, then add half of sweet potatoes. Cook, tossing lightly, until lightly wilted and just tender to bite, 5 to 7 minutes. Transfer to a paper towel-lined plate and set aside. Repeat to cook remaining sweet potatoes.
Meanwhile, put eggs in a small saucepan filled with water. Bring to a boil, then remove from heat and leave eggs in pan 5 minutes. Drain and fill pan with ice water. When cool, peel eggs and cut each in half lengthwise.
Pour broth, ginger, mirin, soy sauce, dashi, and miso into a large pot. Bring to a boil over medium-high heat. Divide broth among four large bowls; add sweet potatoes, eggs, green onions, and bean sprouts to each. Drizzle with chili oil.
* Find mirin and dashi at grocery stores in the Asian-foods aisle or online, and miso in grocery stores' refrigerated foods aisle.