1. Preheat oven to a low broil. Combine first eight ingredients into a food processor or large bowl and, using an immersion blender, blend until coarsely blended.
2. Heat one tablespoon of the oil in a large, heavy nonstick skillet over medium-high heat, and add 3/4 of the tomato puree and cook, stirring, for roughly 10 minutes, until the sauce darkens and thickens. Season to taste with salt.
3. Turn the heat to low, and simmer, stirring often, for about five minutes, until the sauce coats the front and back of a spoon. Taste and adjust salt.
4. Add broken-up pieces of matzah to a large mixing bowl. Cover with the cooked tomato puree. Stir until all pieces of matzah are combined.
5. Return half of the covered matzah to the skillet. Flatten and cover with half of the shredded cheese. Top that with the rest of your covered matzah, cover with the last quarter of your tomato puree and the rest of your cheese.
6. Place the skillet of your cheesy, tomato matzah in the oven and broil until cheese is golden and melted, 4–5 minutes.
7. Meanwhile, pour the last tablespoon of oil into a nonstick skillet to lightly coat. Heat over medium heat. Add eggs and fry until whites are set but yolks are still runny, about 4 minutes.
8. Top chilaquiles with chopped onion, cilantro and lime wedges. Top with fried eggs and serve with remaining sauce alongside.