ut the oats in a food processor and pulse three times to roughly chop. Add half of the beans and pulse into a coarse paste, about 6 pulses. Add the egg, cumin, and 1/2 teaspoons salt and process to mix well, about 1 minute. Transfer the bean mixture to a large bowl. Stir in the remaining beans, the cheese, scallion, and cilantro.
With wet hands, form the bean mixture into four 1/2-inch-thick patties and transfer to a lightly oiled plate. Refrigerate for 10 to 15 minutes to let the burgers set up.
Heat a large heavy-duty skillet (preferably cast iron) on high heat until very hot; add the oil and swirl the pan to coat the bottom. Cook the burgers until browned, with a good crust, 2 to 3 minutes; then carefully flip and cook, flipping again if necessary, until the burgers feel firm when pressed with a fingertip, another 3 to 5 minutes. Serve the burgers in the buns, topped with the salsa and avocado.