Stir together the ground flax seeds and 2 ½ tablespoons of water in a small bowl and set it aside to thicken.
Drain the black beans over a bowl to reserve the liquid.
Add the drained black beans to a food processor and pulse a few times to break them down.
Add the oats and pulse a few times to incorporate them.
Add the thickened flax egg, salt, pepper, and 1 tablespoon of the reserved bean liquid to the food processor, and pulse a few times to get a chunky mixture.
Transfer to a bowl and let the mixture sit for 5 minutes so the oats can absorb the liquid.
Shape the mixture into 2 patties about ¾ inch thick, then refrigerate for 30 minutes.
Heat the olive oil in a cast iron skillet over medium heat.
Add the patties to the skillet and cook for 4 to 5 minutes on the first side.
Flip with a spatula and cook for 4 to 5 minutes on the second side, until firm and charred.
Serve in a bun with your condiments of choice or serve over greens.