In a food processor, combine the fish fillets, green beans, cilantro, red curry paste, fish sauce, lime juice, sugar, salt, and black pepper. Pulse until the mixture is smooth but still slightly chunky.
Transfer the mixture to a bowl and fold in the breadcrumbs until well combined. Cover and refrigerate for at least 30 minutes to firm up.
Once chilled, shape the mixture into small patties, about 2 inches in diameter.
Heat about 1/4 inch of vegetable oil in a large skillet over medium heat. Fry the fish cakes in batches for about 4-5 minutes on each side, or until golden brown and cooked through. Drain on paper towels.
Serve warm with sweet chili sauce or a lime wedge for dipping.