Cut the tortillas into triangles and let sit at room temperature for 1-2 hours.
Set the oven’s broiler to high. Place the halved tomatoes, quartered red onion, halved jalapenos, and garlic cloves on a sheet tray and drizzle with olive oil. Broil for 10 minutes, turning halfway, or until well charred.
Remove the stems and seeds from the dried chiles, then toast them in a skillet over medium heat for 4-5 minutes. Move the chiles to a small pot, add just enough water to cover, then simmer for 10 minutes or until chiles are softened.
Add the roasted vegetables, softened chiles, juice of 1 lime, and a pinch of salt and pepper to a blender or food processor and blend until combined. If the sauce is too thick, add small bits of the water used for softening the chiles until a smooth consistency is achieved.
In a cast iron pan or other wide skillet, pour in enough vegetable oil to cover the bottom of the pan and heat to medium-high. Add tortillas pieces a handful at a time, being careful not to crowd the pan. Fry until golden brown and remove; repeat in small batches until all tortillas have been fried.
Drain the pan if necessary, then add all the fried tortillas back into the pan, turn the head down to medium-low, and pour in the sauce. Toss to cover the tortillas and cook for 2 minutes, until the sauce is heated through.
Sprinkle cheese over the top of the mixture and remove from heat. Place a pile of the Chilaquiles onto plates, top with fried egg and other desired garnishes, and serve immediately.