Season the chicken with salt and pepper. Heat the oil in a large pot over medium-high heat until shimmering. Cook the chicken, skin-side down, until crisp, about 7 minutes. Flip and cook until the second side has browned, about 4 minutes more. Add the broth, adjust the heat to maintain a simmer, cover, and cook until the chicken has cooked through, about 12 minutes more. Meanwhile, mix the butter and miso together in a small bowl.
Transfer the chicken to a plate and shred with a fork and knife. Discard skin and bones. Bring the broth to a hard boil over high heat. Add the noodles and cook according to the package instructions. Drain, reserving broth for another use if desired, and immediately toss in a serving bowl with the miso-butter. Toss to coat, then add in the chicken. Top with scallions, cilantro and sprouts. Serve immediately with lemon wedges
