Cook the onion in a dry skillet with a spoonful of the oil until soft, about 5 minutes. Add the cumin, chili powder and salt, then the chicken, beans and broth.
Simmer 5 to 8 minutes, until the mixture is thick and no liquid runs when you drag a spoon through it. Spread it on a plate and chill it until cold, at least 30 minutes.
Warm the tortillas briefly so they fold without cracking. Lay one flat, put a quarter of the cold filling slightly below the centre, and scatter cheese over it.
Fold the bottom edge up over the filling, fold both sides in tightly, then roll it up into a sealed parcel. Press out any air as you go.
Heat the oil in a wide skillet to 350F to 360F, enough to come about a third of the way up a parcel. A scrap of tortilla dropped in should sizzle steadily and brown in about 40 seconds.
Lower two parcels in seam side down and fry 2 to 3 minutes, until the seam is set and golden, then turn to brown the remaining sides, about 2 minutes more in total.
Lift out with tongs and drain on a rack rather than on paper, so the underside stays crisp. Fry the remaining two.
Rest 2 minutes before cutting, and serve with sour cream, salsa or guacamole.