Cook the noodles according to the package directions, adding the bok choy during the last minute of cooking. Drain and run under cold water to cool.
Transfer the noodles and bok choy to a large bowl. Add the bell pepper, oil, vinegar, Sriracha, and ¼ teaspoon salt and toss to combine.
Meanwhile, heat broiler. In a small bowl, combine the jelly, ketchup, and ginger.
Place the chicken on a broilerproof baking sheet lined with foil. Season with ¼ teaspoon salt.
Broil the chicken until cooked through, 4 to 5 minutes per side, brushing with the jelly mixture twice during the last 3 minutes. Serve with noodles.