Preheat the oven to 450°F (235°C) and line a small baking sheet with parchment paper.
Cut the tomatoes in half and place them cut side up on the prepared baking sheet.
Add the onion to the sheet cut side up along with the unpeeled garlic. Bake for 20 minutes.
Meanwhile, make the chips by heating the vegetable oil (or other neutral oil) in a medium sized pot or pan. Heat for 4 minutes on medium-high.
Line a large plate with paper towels.
Stack the tortillas and quarter them into triangles with a pizza cutter or sharp knife. You should have four stacks of triangles.
Reduce the heat of the oil to medium and add one stack at a time, dropping each chip individually into the oil. Fry for 2 minutes and push down into the oil if needed with a slotted frying strainer. Remove and place on the plate with paper towels. Repeat with the remaining stacks of tortillas and set aside.
By now, the onion, garlic, and tomato should be done so remove them from the oven and add them to an immersion blender or high-speed blender.
Add the chicken broth, olive oil, salt, Mexican oregano, and peppers. Blend until smooth and add to a large frying pan.*
Heat the sauce on medium for 6 minutes then reduce the heat to low and simmer for another 4 minutes.
Add the chips to the pan, tossing until they're coated with sauce then quickly transfer them to a plate. You don't want to leave them too long or they'll get too soggy.
For the eggs, heat a non-stick pan on medium-high until it's hot enough for a drop of water to sizzle. Reduce the heat to medium and add 1 tbsp butter to the pan. Crack each egg into the pan and cook for about 2 minutes.*
Top each plate of chilaquiles with an egg, cheese, sour cream, and cilantro. Enjoy!