Preheat the oven to 450˚. Heat a large ovenproof skillet over medium-high heat and add the olive oil. Season the chicken with salt and pepper and add skin-side down to the skillet. Cook, moving the chicken occasionally, until the skin is browned and crisp, about 10 minutes. Flip the chicken, transfer the skillet to the oven and roast until cooked through, 10 to 12 minutes.
Meanwhile, bring a large pot of salted water to a boil. Stem the kale and thinly slice the leaves; trim and thinly slice the Brussels sprouts. Toss the kale and Brussels sprouts in a large bowl with 1 tablespoon vinegar and a big pinch of salt; set aside until softened, about 5 minutes.
Add the noodles to the boiling water and cook as the label directs. Drain and run under cold water until cool, then add to the vegetables.
Whisk the remaining 1 tablespoon vinegar, the pickled ginger and brine, sesame oil, soy sauce, garlic, a pinch of salt and a few grinds of pepper in a small bowl. Drizzle the dressing over the noodle mixture; toss well. Add half of the sesame seeds and toss; season with salt and pepper. Divide the noodle mixture and chicken among plates; sprinkle with the remaining sesame seeds.