For the fish cakes: In a food processor, pulse the cod until it just forms a paste, 10 to 12 times. The fish should be mostly smooth in texture but can have small chunks. Transfer the fish to a large bowl and add the long beans, curry paste, fish sauce, kaffir lime leaves, garlic, egg and ginger and mix well with your hands. Using 2 tablespoons per cake, form the fish mixture into small cakes about 2 inches wide and 1/2-inch thick. Place the cakes onto a parchment-lined baking sheet and refrigerate for 30 minutes.
For the pickled cucumbers: Combine the white vinegar, 1/4 cup water, the fish sauce, sugar and chiles in a mixing bowl and stir until the sugar is dissolved. Add the diced cucumbers and stir to combine. Refrigerate while you fry the fish cakes.
Heat 1/2-inch vegetable oil in a large skillet over medium heat. The oil is hot enough when you place the tip of a wooden chopstick or the back of a wooden spoon into it and small bubbles form around it. Fry the cakes in batches until light golden brown and cooked through, turning occasionally, from 1 to 2 minutes per side. Place the cooked fish cakes on a baking sheet lined with paper towels to catch any excess oil.
Transfer the cakes to a serving plate and top with pickled cucumbers.