Stir together soy sauce, sesame oil, doubanjiang, and rice vinegar in a small bowl; set aside.
Heat 1 tablespoon neutral oil in a large wok over high until shimmering and fragrant. Add onion, ginger, and garlic; stir-fry until mixture is fragrant and onion begins to soften, about 30 seconds. Transfer mixture to a small bowl; set aside. Do not wipe wok clean.
Add remaining 1 tablespoon neutral oil to wok over high. Add mushrooms; stir-fry until softened, about 1 minute and 30 seconds. Add cooked noodles, soy sauce mixture, and cooked onion mixture; cook, tossing and stirring constantly, until noodles have absorbed sauce, about 2 minutes. Remove from heat. Transfer to a large serving dish; top with scallions and, if desired, cilantro. Garnish with furikake.