Heat the peanut oil in a large pot or Dutch oven over medium-high heat. Add the fresh ginger and all but 1/4 cup of the red onion and cook, stirring occasionally, until just tender, about 5 minutes. Add the ground chicken, turmeric, chili-garlic sauce and 1 teaspoon salt; cook, breaking up the meat, until cooked through, about 5 minutes. Add 1 cup water, the chicken broth and mustard greens (the pot will be very full, but the greens will wilt quickly). Bring to a boil, then reduce to a simmer and cook until the broth has reduced slightly and the greens are tender, about 5 minutes. Season with salt.
Meanwhile, bring a large pot of water to a boil; remove from the heat and add the vermicelli noodles, stirring to separate them. Let the noodles soak until tender, 2 to 4 minutes. Drain, run under cold water and shake dry. Use kitchen shears to cut the noodles into smaller pieces.
Divide the pickled ginger, noodles and chicken-broth mixture among bowls. Top with peanuts, cilantro and the reserved red onion; season with salt.