Preheat your oven to 350 degrees with the rack in the middle position. Spray you cookie sheet with oil.
Preheat a cast iron pan over medium-high. Add vegetable oil, then fry the corn tortillas on each side until starting to crisp. Set aside.
Add the diced onions to the pan and cook, stirring occasionally until translucent. Add the ground beef, chili powder, Mexican oregano, salt, and pepper, and cook until the beef is starting to brown, but pink in some spots. Add the green chiles and garlic, and cook an additional minute.
Coat two tortillas with enchilada sauce, then place one on each side of the pan, leaving space between them, Add a scoop of the beef mixture to each, top with cheese, then another enchilada sauce dipped tortilla. Keep building layers until you've used all of the tortillas up. Top with any remaining meat mixture you have, ladle enough enchilada sauce to cover, then top with the remaining cheese.
Place your enchiladas in the oven for 15 minutes, or until the cheese is bubbling.
While the enchiladas are baking, add a couple of tablespoons of butter and a couple of tablespoons of vegetable oil to a frying pan, and preheat over medium heat. Carefully add a couple of eggs, without breaking the yolks. When the whites of the eggs start turning opaque, cover with a lid and cook until the whites are fully set, but the yolks are runny.
Remove the enchiladas from the oven and carefully slide a thin spatula under each stack to transfer to a plate (a thin metal spatula is very helpful here). Place fried eggs on top, then whichever garnishes you choose.