Heat 1 inch of vegetable oil over medium heat in a cast-iron skillet until it registers 350 degrees F on a candy/frying thermometer. (If you don't have a thermometer, carefully dip the corner of a tortilla wedge into the oil. If it sizzles, you are ready to fry.)
Carefully drop the tortilla quarters into the hot oil and fry, turning them halfway through with tongs or a spider, until crisp but not yet browned, 1 to 1 1/2 minutes. Remove the tortilla quarters to a paper towel-lined plate to drain; set aside.
Heat the enchilada sauce in a small pan over medium-high heat until hot. While it is heating, fry the eggs. Melt the butter in a small skillet over medium heat. Crack the eggs into the skillet and fry until the edges are nice and crispy, a minute or two. Season with salt and pepper.
While the eggs are frying, assembled your chilaquiles. In a shallow 1-quart dish, arrange half of the fried tortillas in a layer. Sprinkle them with half of the cheese. Pour enough of the hot enchilada sauce over the tortillas and cheese to cover most of the chips. Add the remaining chips to the dish in an even layer. Top with the remaining cheese and more sauce. The amount you add here is up to you. You may end up with some leftover enchilada sauce depending on your preference.
Top with the fried eggs and salsa of your choice. Enjoy while it's nice and hot.