In a food processor, pulse the bell pepper, onion, and garlic until finely chopped, 8 to 10 times. Pour the chopped vegetables out on a paper towel-lined baking sheet. Blot with paper towels to remove excess moisture.
Blot the black beans separately on paper towels and place them in a large bowl. Mash the black beans until mostly broken up with a few whole beans remaining. Add the chopped vegetables, breadcrumbs, chipotle chiles, adobo sauce, egg, chili powder, cumin, salt, and black pepper. Stir until well combined. Shape the bean mixture into 4 even patties.
Preheat a grill or large skillet over medium-high. Oil the grill grates or add oil to the skillet and tilt the skillet to evenly coat the bottom. Cook bean patties, in batches if needed, until browned and heated through, 3 to 4 minutes on each side. If topping burgers with cheese, place 1 slice of cheese on each patty during the last minute of cooking.
Serve patties on hamburger buns with desired toppings.
