Combine 1 cup peanut sauce, ginger, garlic and 1 teaspoon soy sauce in a bowl. Add the chicken and mix until the chicken is coated. Place in refrigerator to marinate for 30 minutes.
Heat 1 teaspoon peanut oil in a wok or frypan over medium heat. Cook the snow peas, bean sprouts and spring onions in the oil until heated but still crisp, 3 to 4 minutes. Transfer to a large bowl. Mix in the watercress and coriander. Use a vegetable peeler to add long slices of carrot into the watercress mixture. Drizzle 1 teaspoon soy sauce into the watercress mixture; toss to coat.
Heat 1 teaspoon of oil in the wok or frypan. Cook the marinated chicken until no longer pink inside, about 10 minutes.
Fill a large bowl with hot water. Dip wrappers one at a time into the water for about 2 seconds each. As wrappers are removed from the water, fill each with 2 large spoonfuls of the chicken and a small handful of the watercress mixture. Fold in two opposite ends of the wrapper to meet the filling. Then fold the bottom of the wrapper over the top of the filling and roll. Serve with 1/2 cup peanut sauce for dipping.