In the medium bowl, mash the beans with a fork. Stir in the cheese, tortilla chips, green onions, cilantro, cumin, and egg white. Make sure the ingredients are thoroughly combined. Make 4 patties about ½-inch thick, pressing firmly to make sure the ingredients stick together.
Heat the oil in the skillet over medium-high heat. Cook the patties until browned and heated through, about 3 minutes on each side. Drain on a paper towel.
Serve in gluten-free pita halves with guacamole and Mexican slaw.