To make Mustard Paste: Mix mustard and water until smooth. Stir in sesame oil to make a smooth, shiny paste.
To assemble: Remove bones and skin from warm chicken and shred meat by hand or with cleaver. Place chicken on one side of large shallow mixing bowl. Place lettuce, carrots, cilantro, and green onions on other side of bowl. Drizzle Mustard Paste down sides of bowl around ingredients. Just before serving, toss chicken and lettuce mixture together, mixing in Mustard Paste to distribute it evenly. Add rice sticks, peanuts, and sesame seed; toss lightly. Divide salad equally among 24 plates. Spoon generous 2 tablespoons Soy Dressing over each serving.
To make Soy Dressing: In bowl, combine 2 cups hot chicken broth, 1 1/2 cups soy sauce, 1/2 cup sugar, 1/4 cup distilled white vinegar, 1 tablespoon minced ginger and 1 tablespoon minced garlic. Mix until sugar is dissolved. Cover and refrigerate. (This can be done a few hours ahead to allow the flavors to develop.) Makes about 4 cups.