Combine 1 cup peanut sauce, ginger, garlic, and 1 teaspoon soy sauce in a bowl. Add chicken and mix until coated. Cover and refrigerate for 30 minutes.

Heat 1 teaspoon peanut oil in a wok over medium heat. Add bean sprouts, snow peas, and green onions and cook until heated but still crisp, 3 to 4 minutes. Remove from the heat.

Transfer bean sprout mixture to a large bowl; mix in watercress and cilantro. Use a vegetable peeler to peel long strips carrot into the bowl. Add remaining 1 teaspoon soy sauce and toss to coat.

Heat remaining 1 teaspoon peanut oil in the wok over medium heat. Add marinated chicken; cook and stir until no longer pink in the center and the juices run clear, about 10 minutes.

Fill a large bowl with hot water. Submerge one wrapper in the bowl and hold until soft and pliable, about 2 seconds. Lift wrapper, let excess liquid drip off, and place onto a work surface. Place two large spoonfuls chicken and a small handful vegetable mixture into the center of the wrapper. Fold in the two sides, then fold the bottom up and over the filling; roll to seal. Repeat to make remaining spring rolls.

Serve with remaining 1/2 cup peanut sauce for dipping.