Early in the day or the night before, in a small bowl, combine marinade/dressing ingredients.
Put chicken breasts in a Ziploc bag. Pour 1/2 cup of the dressing/marinade into the bag of chicken and seal, squeezing out excess air. Massage the bag to coat the chicken and refrigerate for at least 6 hours. Refrigerate the remaining dressing.
To prepare salad:
Preheat grill to medium heat. Remove chicken from marinade and wipe excess marinade off chicken pieces with paper towels (discard towels). Grill until no longer pink.
While chicken is grilling, cook vermicelli noodles according to package directions. Drain and rinse under cold water; set aside.
Place 2 cups lettuce on each dinner plate. Top with cooked noodles, carrots, green onions, chicken, cilantro and chopped nuts. Drizzle remaining dressing over each salad.