Chop cherry tomatoes, olives, onion, and basil and add olive oil and balsamic vinegar. Season with salt and keep aside. Place in the food processor 300 grams. of cooked beans, the egg, the parmesan, the garlic, 1/3 cup breadcrumbs, and fresh basil and pulse until you have a homogeneous mixture.
In a bowl, coarsely mash with a fork the remaining 100 g. beans and add the mixture from the food processor. Mix until you have a fairly stiff dough. Pour the rest of the breadcrumbs on a plate. Wet your hands with some olive oil and form 4 large patties. Coat with breadcrumbs and refrigerate for 10 minutes. Heat your skillet on high heat until hot. Pour 1-2 tablespoons olive oil, lower the heat and fry the patties, turning them from both sides for about 5-6 minutes, until golden brown on the surface.
Slightly heat the buns and cut them in half. Place on the bottom bun a lettuce leaf, then the patties with beans, and finally the sauce with fresh cherry tomatoes. Cover with the top of the bun and Enjoy!