Fish Cakes: Place the fish and curry paste in a food processor. Pulse until the fish is blended and forms a paste. Or alternately smash in a mortar and pestle until a paste is formed.
Place the fish paste in a bowl. Mix in egg, lime leaves, green beans, sugar, fish sauce, salt and pepper.
Mix the paste well to work air into the paste. This will make the cakes fluffy after cooking.
Sauce: Dissolve the sugar and salt in the vinegar. Add the chilli flakes, shallots and cucumber then let marinate at room temperature for at least 10 minutes.
To Cook: Heat oil in a heavy frying pan. Test the temperature of the oil by placing a small piece of fish paste in the oil. If the fish paste sizzles then the oil is ready. If the oil begins smoking it is too hot; let it cool off.
Fry a small piece of fish cake and taste. Adjust seasoning with salt and/or chilli, if necessary.
Take 3 tablespoons of fish paste and flatten into a cake. Lower gently into the hot oil. Continue until all the paste is gone. You may have to fry in batches depending on the size of your frying pan. Fry cakes until golden brown. Gently remove and drain on paper towel.
Place on a platter and serve the sauce in a small bowl for dipping or spooning.