Heat the olive oil in a large skillet over medium heat.
Add half of the black beans to a large mixing bowl and lightly mash with a potato masher, keeping the mixture chunky. Add the remaining whole beans. Stir in the scallions, corn, salsa, barbecue sauce, smoked paprika, garlic powder, and cilantro. Season with salt and black pepper to taste.
Add the breadcrumbs, starting with 7 tablespoons, and mix until the mixture holds together. If needed, add 1 to 2 additional tablespoons until firm enough to shape.
Divide the mixture into 4 portions and form into burger patties using lightly damp hands.
Pan-fry the patties in the hot skillet for 4 to 5 minutes per side, until golden brown and lightly crisp.
Serve the patties on hamburger buns with toppings.