To prepare dumplings, place first 4 ingredients in a food processor; pulse 8 times or until finely chopped. Add ginger, oil, salt, and scallops; pulse 4 times or until scallops are finely chopped. Spoon mixture into a medium bowl; add crabmeat and egg, tossing to combine.
Working with 1 gyoza skin at a time (cover remaining skins to prevent drying), spoon about 1 tablespoon filling into center of each skin. Gather up edges of skin around filling; lightly squeeze skin to adhere to filling, leaving top of dumpling open. Smooth surface of filling with the back of a spoon dipped in water. Place dumplings on a baking sheet sprinkled with cornstarch; cover loosely with a damp towel to prevent drying.
Line each tier of a 2-tiered (10-inch) bamboo steamer with 4 of the lettuce leaves. Arrange 9 dumplings, 1 inch apart, over lettuce in each steamer basket. Stack tiers, and cover with the steamer lid. Add water to a large skillet to a depth of 1 inch; bring to a boil. Place steamer in pan; steam dumplings 10 minutes. Remove dumplings from steamer.
To prepare sauce, combine sugar and remaining ingredients, stirring with a whisk until sugar dissolves. Serve sauce with dumplings.