Turn on the broiler and position the rack 4 inches below the heat. Spread chicken pieces on a rimmed baking sheet in a single layer and coat with the vegetable oil, sprinkle with salt and pepper. Sprinkle on both sides with the sesame seeds. Broil, turning thighs once or twice to prevent seeds from burning, until chicken is cooked through and golden in places and the pan is almost dry, 10 to 15 minutes total.
Add noodles to the boiling water, cook for 3 minutes, stirring occasionally, then start tasting when the noodles are pliable but not quite ready. Add the broccoli and cook until it turns bright green, another minute or two. Drain noodles and broccoli, rinse with cold running water to stop the cooking, then shake the colander and drain again.
Slice chicken into narrow strips and put in a large serving bowl with any pan juices and the green onions, soy sauce and sesame oil. Toss to coat, add the noodle mixture and toss again. Taste, adjust seasoning and serve warm or at room temperature.