Heat a nonstick skillet over medium heat and add the chorizo (with casings removed). Cook, breaking up the meat with a spatula, until browned and crumbly, about 5–7 minutes. Use a slotted spoon to transfer the chorizo to a plate, leaving the rendered fat in the skillet.
In a mixing bowl, whisk together the eggs, milk, and salt and pepper to taste until combined.
Return the skillet with the chorizo fat to medium heat and add the egg mixture. Cook, gently scraping and folding the eggs, until they are just set but still moist, about 3–4 minutes.
Add the cooked chorizo back to the skillet with the scrambled eggs and stir to combine. Sprinkle the shredded cheddar over the mixture and allow it to melt, about 1 minute.
Warm the flour tortillas in a separate skillet over low heat or briefly in the microwave until pliable. Divide the chorizo-and-egg mixture evenly among the tortillas.
Top each portion with fresh salsa and sliced avocado if using, then roll up the tortillas, folding in the edges to enclose the filling. Serve immediately while hot.