To prepare marinade, combine first 9 ingredients in a large heavy-duty zip-top plastic bag; add chicken. Seal bag; marinate in refrigerator overnight.
Remove chicken from bag; discard marinade. Let stand at room temperature 30 minutes.
While chicken stands, cook noodles according to package directions, omitting salt and fat. Drain.
To prepare sauce, combine 1/4 cup rice vinegar and next 6 ingredients (through chile paste) in a small bowl.
Preheat grill to medium-high heat.
Place chicken on grill rack coated with cooking spray. Cook 6 minutes on each side or until done. Let stand 5 minutes. Cut chicken crosswise into slices.
Place 3/4 cup noodles and slices from 1 chicken breast half in each of 6 bowls. Top each serving with 1/3 cup carrot, about 3 tablespoons cucumber, 2 teaspoons basil, and 1 teaspoon peanuts. Drizzle each serving with 1 1/2 teaspoons sauce, and serve with 1 lime wedge.
Tips: To make carrot ribbons, use a vegetable peeler.