Add potato cubes to a large skillet set over medium-high heat. Cook, without stirring, for about 5 minutes or until bottoms start to brown. Stir to flip and continue to cook, stirring every 3 to 4 minutes, until potatoes are tender and browned on most sides, about 12 to 15 minutes more.
Meanwhile, cook chorizo in a skillet until browned and starting to crisp, breaking up large chunks into smaller pieces with a wooden spoon.
Wrap tortillas in foil and place in the oven until warmed through.
Whisk 3 eggs together in a small bowl.
Heat an 8-inch skillet over medium heat. Add a teaspoon or so of oil. Add half of egg mixture, swirling to spread egg out to the edges of the skillet.
Cook for 30 seconds until bubbling, then sprinkle with 1/4 cup or so of shredded cheese. When melted, place a tortilla on top of the whole thing and press lightly to adhere. At this point, the egg should be cooked through.
Flip to invert the whole thing, tortilla side down, onto a work surface. Spoon a bit of potato and chorizo down the middle of the tortilla. Top with pico de gallo and a drizzle of sour cream as desired.
Fold in edges and roll. Wrap in parchment or foil as desired (keep foil-wrapped burritos in a warm oven while you prepare the others).
Whisk remaining 3 eggs together and repeat. Enjoy warm.