In a skillet over medium heat, add the olive oil. Once hot, add the chorizo sausage, breaking it apart with a spatula. Cook for about 5-7 minutes, until the chorizo is browned and cooked through.
While the chorizo is cooking, crack the eggs into a bowl and add the milk, salt, and pepper. Whisk until well combined. Once the chorizo is cooked, reduce the heat to low and pour the egg mixture into the skillet. Gently stir the eggs with the spatula, allowing them to cook slowly until they are fluffy and just set, about 3-4 minutes.
Sprinkle the shredded cheddar cheese over the eggs and chorizo mixture. Stir gently until the cheese melts and becomes gooey, creating a delicious filling.
Warm the flour tortillas in a separate skillet or microwave until soft and pliable. Spoon an equal amount of the chorizo and egg mixture onto each tortilla. Fold the sides in, then roll the tortilla from the bottom up to enclose the filling.
Serve the burritos warm, topped with fresh salsa and optional avocado slices. Enjoy your Chorizo Breakfast Burrito immediately for the best flavor and texture.