Pulse 1 can beans in a food processor with mayonnaise, bread crumbs, cumin, oregano, and cayenne until a coarse purée forms.
Transfer to a bowl and stir in cilantro and remaining can beans.
Form mixture into 4 patties.
Heat oil in a 12-inch heavy skillet over medium-high heat until it shimmers.
Cook burgers until outsides are crisp and lightly browned, turning once, about 5 minutes total.
Serve on buns.
