Cook noodles according to package directions, omitting salt and fat. Drain and rinse under cold water. Drain well.
While noodles cook, combine rice vinegar and next 5 ingredients (rice vinegar through sesame oil) in a large bowl.
Add noodles and chicken to vinegar mixture; toss to coat. Arrange 1 cup spinach and 1/4 cup carrot on each of 4 plates. Top each serving with 1 1/2 cups chicken mixture; sprinkle each serving with 1 tablespoon peanuts.