Cook the noodles until just tender; rinse under cold water; drain, and toss with 1 teaspoon sesame oil. Arrange the noodles in a large serving bowl.
Arrange the vegetables in concentric circles over the noodles and then pile the chicken in the center. Sprinkle the scallions on top.
Serve at room temperature or chilled, with the Chinese peanut dressing.
For Chinese Peanut Dressing: In a food processor fitted with the metal blade or in a blender, finely chop the ginger and garlic. Add the remaining ingredients in the order listed and process until smooth. The dressing should be the consistency of heavy cream. If it is too thick, add more water or chicken broth; if too thin, add more peanut butter. Refrigerated, in a covered container, the dressing will keep for 2 to 3 weeks.