Begin by frying the onion in oil until soft, about 7 minutes.
Drain, but do not rinse, the black beans. Place them in a bowl and use a fork to mash them. Keep mashing until they're mostly broken up, but still have some whole beans visible. Add the breadcrumbs, onion, egg, chili powder, oregano, salt, and pepper. Stir until everything is combined, then let the mixture sit for 5 minutes.
Heat a tablespoon or two of olive oil with an equal amount of butter in a skillet over medium-low heat. Form the bean mixture into patties the same size as the buns. Place the patties in the skillet and cook them about 5 minutes on the first side. Flip them to the other side, place 2 slices of cheese onto each patty, and continue cooking them for another 5 minutes, or until the burgers are heated through.
Toast the buns, then spread with mayonnaise. Place the patties on the buns. Top with lettuce and tomato, and serve hot!