Cook the noodles according to the package directions. Drain and run under cold water to cool.
Meanwhile, season the chicken with ½ teaspoon salt and ¼ teaspoon pepper.
Heat 3 tablespoons of the oil in a skillet over medium-high heat.
Cook the chicken in batches until golden brown and cooked through, about 2 minutes per side. Cut into strips.
In a large bowl, whisk together the vinegar, soy sauce, ginger, and remaining 3 tablespoons of oil. Add the noodles, chicken, cabbage, and scallions and toss.