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Heat a large nonstick skillet over medium-low heat. Add the chorizo and cook, stirring, 8 to 10 minutes, until the sausage is golden brown and cooked through. (Be careful when stirring so that the grease from the chorizo doesn't pop out of the pan.) Remove the chorizo with a slotted spoon and drain on a paper towel-lined plate. Set aside. Reserve 1 1/2 tablespoons of the drippings from the sausage. Cook's Note: You can also opt to cook half the chorizo now, remove some of the rendered fat, cook half the eggs in the same skillet and remove when finished. Repeat with the remaining ingredients.
Heat a clean nonstick skillet over medium-low heat. Add 1/2 teaspoon of oil to the pan and place 1 tortilla in the oil. Cook for 1 minute and turn. Cook an additional 30 seconds. The tortilla will be slightly golden but still soft. Transfer the warm tortilla to a plate and cover with foil to keep warm. Continue to cook the remaining tortillas, using about 1/2 teaspoon of oil for each tortilla. Cook's Note: You can also opt to deep-fry the tortillas as well.
In the same skillet, heat half of the reserved drippings from the chorizo over medium-high heat. One at a time, carefully break 3 eggs into the skillet, making sure that the edges of the eggs don't touch. Season the eggs with 1/8 teaspoon salt each, and fry each egg, about 1 minute. Using a spatula, carefully flip the eggs and fry for 1 minute longer, or to the desired degree of doneness. Transfer the eggs to a plate and sprinkle each egg with 2 tablespoons of the grated cheese.
Using the remaining chorizo drippings, cook the remaining 3 eggs and sprinkle each with 2 tablespoons of cheese.
To serve, place 1 tortilla on each plate. Top each tortilla with a fried egg and cheese. Sprinkle chorizo over the egg and top with salsa. Serve warm.