Add the shrimp to a food processor along with the mayo, egg, green onions, garlic, ginger, salt, and pepper. Process until the mixture is mostly smooth, but still has a few chunks here and there. Pre-heat a large non-stick pan over medium-high heat with 1-2 teaspoons of oil. Once hot, add 3-4 tablespoons of shrimp mixture to your hand and place it in the pan. Carefully press the mixture into a cake. Repeat with a couple more shrimp cakes making sure not to over-crowd the pan. Cook until golden brown, about 3-4 minutes, flip and cook another 3 minutes. Repeat with remaining shrimp mixture and keep cooked cakes warm in the oven.