In a large pan we heat the oil. We get ready a paper-towel lined dinner plate.
When the oil is hot enough to sizzle a tortilla edge, we quick-fry a tortilla. One at a time, for about 3 seconds a side, we soften the tortillas. We drain them on the paper towels and then wrap in aluminum foil and keep warm in the oven.
We pour about ½ tablespoon of the oil in the frying pan into a non-stick pan and heat over medium-low heat. When hot, we crack the eggs on the side of a bowl and gently open the eggs in the non-stick pan. We cook, covered with a loose draping of aluminum foil, for a few minutes. When the eggs are cooked to the desired doneness, we top with a little kosher salt.
We remove the tortillas from the oven. We place one on each plate and then top with two sunny-side-up eggs. We spoon on the green salsa, leaving the yolks of the eggs exposed. We top with a bit more salt, and some cilantro and cheese.