To prepare dressing, combine the first 9 ingredients, stirring with a whisk, and set aside.
To prepare salad, cook carrots and noodles in boiling water 1 minute; drain. Place noodle mixture in a large bowl. Add bok choy and next 5 ingredients (through mint); toss gently to combine. Pour dressing over noodle mixture; sprinkle with 2 tablespoons peanuts. Toss gently to combine. Place 1 lettuce leaf on each of 6 plates. Top each leaf with 1 cup salad; sprinkle each serving with 1/2 teaspoon peanuts.