Whirl cream cheese, ginger, garlic, and soy sauce together in a food processor until smooth. Transfer mixture to a medium bowl. Stir in green onions and crab.
Whisk together egg and 1 tbsp. water in a small dish. Lay wonton wrappers flat. Brush with egg wash, covering completely with a thin layer. Spoon a scant 1 tbsp. crab mixture onto center of each wonton. Pull up corners so all four meet in the center, pressing edges together to seal. Set on a rimmed baking sheet lined with a sheet of parchment paper.
Fill a wide pot with 2 in. oil. Heat over medium heat to 350° on a deep-fry thermometer. Working in batches of 3 or 4, fry rangoons, turning as needed, until golden brown, about 8 minutes. Use a slotted spoon to transfer rangoons to a plate lined with paper towels. Let cool slightly before serving with dipping sauces.
Make ahead: Through step 2, up to 3 hours, chilled, or 2 weeks, frozen.
Note: Nutritional analysis is per rangoon.