Make the dipping sauce: Stir juices, ginger, oil, sugar, and chile in a bowl. Refrigerate until ready to use, up to 1 day.
Make the shrimp rolls: Remove 6 of the best whole leaves from each cabbage; set aside. (Reserve remaining cabbage for another use.) Place shrimp in a medium bowl; sprinkle with salt. Stir in water chestnuts, ginger, scallions, cilantro, garlic, and chile. Add egg white and cornstarch; stir until combined. Turn mixture out onto a cutting board. Finely chop all ingredients just until a paste forms.
Place 2 tablespoons filling at base of 1 cabbage leaf; shape into a log. Fold leaf's ends in and roll tightly around filling. Place in a bamboo steamer. Repeat with remaining leaves and filling.
Fill a wok or large skillet with 1 inch of water; bring to a boil. Place steamer over wok (bottom slats should not touch water); reduce heat to medium. Cover, and steam until cabbage leaves are soft and shrimp are cooked through, 20 to 25 minutes. Halve rolls diagonally, and serve with the dipping sauce.