Place the black beans in the bowl of a food processor. Add the onion, cilantro, garlic, salt, and chipotle and process until the mixture is the texture of very coarse cornmeal. Fold the sweet potato in by hand, just to combine.
Divide into eight portions (a heaping ⅓ cup apiece) and form by hand into patties about ½ inch thick. Cover in plastic wrap and refrigerate for at least 2 hours or overnight. (You can refrigerate them in an airtight container for up to 1 week.)
To fry the patties, heat the oil to a depth of ¼ inch in a large skillet over medium-high heat. Once it shimmers, add as many patties as will fit without crowding and cook until crisp and deeply browned, about 5 minutes per side. Transfer to paper-towel-lined plates to drain. Add more oil if needed and repeat with the remaining patties.
Serve on the burger buns, layered with the aioli and other condiments of your choice and with the lettuce, avocado, onion, and tomato.