In a food processor, pulse the 1/4 cup basil and the garlic until finely chopped. Add the Parmigiano, 1/2 cup of the beans, 1/4 cup of the panko, the egg, and 1/2 tsp. salt and process until smooth. Put the remaining beans in a large bowl and coarsely mash with a fork. Add the puréed bean and basil mixture and mix well.
Spread the remaining 1/4 cup of panko on a plate. With wet hands, form the bean mixture into four 3/4-inch-thick patties. Coat each patty with the panko, transfer to a plate, cover, and refrigerate for 10 minutes.
Meanwhile, in a small bowl, combine the tomato, olives, onion, balsamic vinegar, and the remaining 12 basil leaves. Set aside.
Heat a heavy-duty 12-inch skillet on high heat until very hot, about 1 minute. Add the oil and swirl to coat the bottom. Cook the burgers, flipping once, until browned, 5 to 8 minutes total. Serve the burgers on the buns, topped with the tomato-olive relish.