Heat the butter in a small skillet over medium heat. Add the onion, bell pepper, and garlic and cook, stirring occasionally, until softened. In a bowl, mash 1/3 of the beans with a fork and stir in the remaining beans, onion mixture, bread crumbs, and cilantro and season with salt and pepper. Stir in the eggs.
Form bean mixture into 8 (4-inch) patties (mixture will be wet and soft.). Gently dredge patties in cornmeal and transfer to a large plate lined with plastic wrap. Refrigerate the cakes for 30 minutes and up to 12 hours.
Pour enough oil into a heavy skillet to reach 1/4-inch up the side. Heat the oil over medium heat until hot but not smoking. Working in batches, cook the patties, turning once, until well browned, about 1 1/2 minutes per side. Transfer the cakes to paper towels to drain.
Divide the cakes among plates and serve with the sauce.
Stir together cilantro, vegetable oil, soy sauce, garlic, lime juice, and chile. Season with salt and pepper. Yield: about 1 cup