Step 1

Heat the oil in a large nonstick frying pan or cast iron skillet (12 inches or larger) over medium heat until shimmering. Add the chorizo and cook, stirring frequently and breaking it up into small pieces, until cooked through, about 10 minutes.

Step 2

Stir in the potatoes and cook until warmed through and browned, about 5 minutes more. Transfer the mixture to a large bowl and set aside. Wipe out the pan with paper towels.

Step 3

Whisk the eggs in a large bowl until the whites and yolks are completely mixed and the eggs are a bit frothy. Whisk in the milk and salt until just combined.

Step 4

Heat the butter in the same pan over medium-low heat until melted. Pour in the egg mixture. (Do this in batches if the pan is too full.) Let sit undisturbed until the eggs just start to set around the edges, about 2 minutes. Using a rubber spatula, push the set eggs from the edges into the center. Spread the uncooked eggs back into an even layer. Repeat, pushing the set eggs from the edges into the center every 30 seconds until almost set, for a total cooking time of 8 to 10 minutes. (The top of the eggs should still be slightly wet.)

Step 5

Remove the pan from the heat. If serving the burritos immediately, proceed to assembling them. If not serving immediately, allow the chorizo, potatoes, and eggs to cool completely before assembling.

Step 6

Assemble the burritos: Stack the tortillas on a microwave-safe plate and cover with a damp paper towel. Microwave until soft and pliable, about 30 seconds.

Step 7

Place the tortillas on a work surface. Sprinkle 1/4 cup of cheese across a tortilla, just below the center. Top the cheese evenly with 1/2 cup of the chorizo and potato mixture, then top with 1/2 cup of the scrambled eggs. Roll the burrito tightly by folding the sides over the filling, then rolling from the bottom up. Repeat with assembling the remaining burritos.

Step 8

Reheating: If not serving immediately, wrap each burrito tightly in foil. Refrigerate or freeze in resealable plastic bags. Reheat uncovered in a 325°F regular or toaster oven until warmed through, about 20 minutes if refrigerated or about 30 minutes if frozen. Microwaving is not recommended, as the burritos will be sogg