In a medium bowl, whisk together the eggs and milk and season with salt and pepper.
Heat 1 tablespoon of the oil in a medium skillet over medium heat. Add the chorizo and cook, stirring, until just beginning to brown, about 5 minutes. Scoop the chorizo out of the skillet with a slotted spoon and transfer to a plate. Wipe the skillet out with a paper towel.
Return the skillet to medium to medium-low heat and heat the remaining 1 tablespoon oil. Add the egg mixture and cook, stirring, about 3 minutes. Add the chorizo and cook, stirring, about 2 to 4 minutes more depending on the desired firmness of the eggs. Serve immediately.
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