Combine in a food processor the ginger, shallots, carrot, green onion, cilantro, peanut oil and lemon juice. Pulse until fine. Put vegetable mixture in a mixing bowl, add the mayonnaise and the crabmeat and season with salt and pepper. Be careful not to mash the crabmeat, you want that texture when you bite into the wonton.
Lay a wonton wrapper on a flat surface and brush with the beaten egg white. Drop 1 tablespoon of the crab filling onto the center of the wrapper. Fold the wonton in half, corner to corner, to form a triangle. Press around the filling to knock out any air bubbles, then press the seam together to seal so the filling doesn't seep out. You can leave them this shape or continue on by brushing the 2 side points with beaten egg white. Lay your index finger in the center so you have something to press up against, then fold the 2 sides into the center, slightly overlapping, and press the dough against your finger with your thumb to form a tight seal. Lightly dust the filled wontons with cornstarch to keep them from sticking together and place them on a cookie sheet. (When these are folded they look like Pope hats.)
Heat 2 to 3 inches of oil in a deep heavy saucepan to 370 degrees F on a deep-fry thermometer. Add a few of the wontons to the oil and cook, turning them 3 or 4 times to get them nicely browned all over. Carefully lift them out of the pan with a slotted spoon and onto a paper towel-lined platter to drain. Keep going to cook all of the wontons.
Stir together the dipping sauce and serve with the wontons.