Drain and rinse beans. Dice bell pepper, chop cilantro, and slice onion. Set aside separately.
Place 2 1/2 cups beans in a large bowl and mash with a potato masher or fork until mostly smooth.
In a separate bowl, combine the eggs, 1 tablespoon of the oil, cumin, and salt. Stir to blend.
Add the egg mixture, bread crumbs, remaining beans, bell pepper, cilantro, and onion to the bowl with the mashed beans. Stir together until evenly combined.
Divide the mixture into 6 equal portions, about ½ cup each. Lightly pack into 1-inch thick patties.
Heat 1 tablespoon of the oil in a large skillet over medium heat until shimmering.
Carefully place patties in the skillet and cook until well browned on both sides, about 8-10 minutes total. Remove from heat and repeat with remaining patties, adding more oil as needed.
Mash the avocados and spread onto buns; add a bean burger to each.
Serve sliced apples on the side.