To make the soup: Remove the claws from the crabs and quarter the bodies. In a large stockpot, add the oil, ginger, shallots, and chile for 2 minutes until fragrant. Then add the crabs and miso. Cover with 1 gallon of cold water and place over medium heat; simmer for 30 minutes. In the meantime, you can focus on assembling the wontons.
Heat a skillet over medium flame and coat with the peanut oil; add the ginger, shallot, and carrot, saute for 2 minutes to soften. Put the lump crabmeat in a mixing bowl and scrape in the ginger mixture. Fold in the green onion, chopped cilantro, mayonnaise, and lemon juice; 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. Brush 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.
When the wontons are all filled and folded, strain the soup broth to remove the solids and pour it into the hot pot. Bring the soup to a simmer. Toss in the shiitake mushrooms and the wontons; simmer for 15 minutes until the wontons are cooked. Add the lime juice and cilantro; ladle into soup bowls and serve.