Heat the oven to 375F and spread about half a cup of the enchilada sauce across the bottom of a 9x13 dish.
Toss the shredded chicken with the diced onion, cumin, salt and half a cup of the sauce, plus about a third of the cheese.
Heat the oil in a small skillet over medium heat. Slip each tortilla into the hot oil for 5 to 10 seconds a side, just until it goes soft and pliable, then drain it on paper towel. Do not let it crisp.
Dip each softened tortilla in the sauce so both faces are coated, lay it flat, add a line of filling down the centre, and roll it up.
Lay the rolls seam side down in the dish, packed snugly in a single layer.
Pour the remaining sauce over the top so every tortilla is covered, then scatter the rest of the cheese over it.
Bake uncovered for 20 to 25 minutes, until the sauce bubbles at the edges and the cheese is melted. Rest 5 minutes before serving so the rolls hold together.