Heat the oil over medium heat in a 10- to 12-inch cast-iron skillet until oil barely shimmers. Heat your broiler to high and place a rack about 6 inches under the burner.
Add tortilla pieces a few at a time to the skillet. Once they're all in, they'll overlap, but move them around and flip them with tongs every now and then and they'll fry evenly.
When the chips are crisp and slightly brown around the edges, ladle in the salsa. If you prefer your chilaquiles on the crunchy side, go with 9 ounces. If you like it soft, as I do, go with 12.
Reduce the heat and allow the salsa to simmer and absorb into the chips a bit, 5 minutes. Since they’re fried, the chips will hold together forming layers.
Arrange the softened tortillas so that you can crack 4 eggs evenly across the surface, one in each quarter of the pan.
Park the pan under the broiler for 3 minutes or until the whites set. The yolks will hopefully still be a bit runny.
Remove skillet and scatter surface with the cotija cheese and the cilantro. Deliver to the table and serve right out of the skillet.