Remove the chorizo from its casings and crumble it into a cold nonstick skillet. Heat the skillet over medium and cook the chorizo, breaking it up with a spatula, until browned and crumbly, about 5–7 minutes. Transfer the cooked chorizo to a bowl and leave the rendered fat in the skillet.
In a medium bowl, whisk together 4 large eggs, 1/4 cup milk, and salt and pepper to taste until combined.
Pour the egg mixture into the skillet with the reserved chorizo fat. Cook over medium heat, gently stirring and folding the eggs, until they are just set but still moist, about 3–4 minutes.
Return the cooked chorizo to the skillet with the eggs and stir to combine. Sprinkle 1 cup shredded cheddar cheese over the mixture and stir briefly until the cheese melts and is evenly distributed. Remove from heat.
Warm the 4 large flour tortillas so they are pliable. To do this on the stovetop, heat a separate skillet over medium, add 1 tablespoon olive oil, and warm each tortilla 15–30 seconds per side until pliable and lightly heated. (If you prefer, you may microwave the tortillas wrapped in a damp paper towel for 20–30 seconds instead.)
Divide the chorizo-and-egg mixture evenly among the warmed tortillas. Top each with fresh salsa and sliced avocados, if using.
Fold in the sides of each tortilla and roll up to form burritos. Serve immediately while hot.