Step 1

Combine dried chiles, garlic, oregano, and 2 cups water in a small saucepan; add more water if needed to cover chiles. Bring to a simmer over medium heat. Cook until chiles are soft, 10–15 minutes.

Step 2

Transfer chiles and cooking liquid to a blender; purée until smooth. Strain red chile sauce through a fine-mesh sieve back into pan; discard solids. Whisk in salt and flour and bring to a simmer over medium heat. Cook, whisking constantly, until thickened, 5–7 minutes.Do ahead: Sauce can be made 1 week ahead. Transfer to an airtight container; cover and chill.

Step 3

Remove chorizo from casings if needed and place in a small saucepan. Add 1 cup red chile sauce. Cook over medium heat, stirring often and breaking up sausage with a spatula, until sausage is cooked through, 8–10 minutes.

Step 4

Meanwhile, peel potato and grate on the large holes of a box grater. Transfer to a fine-mesh sieve and rinse under cold water to remove excess starch; drain well and pat dry. Transfer potato to a medium bowl. Add garlic powder, ¾ tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ¼ tsp. pepper and toss to coat.

Step 5

Heat 3 Tbsp. oil in a medium nonstick skillet over medium-high. Add potato, pressing gently. Cook, undisturbed, until a golden brown crust forms, 5–7 minutes. Break up potato with a spatula and continue to cook, stirring occasionally, until golden brown all over, 6–8 minutes more (add more oil if pan seems dry). Transfer hash browns to a plate.

Step 6

Return skillet to medium heat. Add eggs and butter; season with salt and pepper. Cook, stirring with a heatproof spatula, until eggs are glossy and just set, about 3 minutes. Scrape eggs onto a plate and wipe out pan.

Step 7

To assemble burritos, place tortillas on a work surface. Arrange half of hash browns, scrambled eggs, cheese, and chorizo mixture in parallel strips across center of each tortilla, leaving a 1" border on left and right sides. Spoon half of green chiles over each. Fold in sides of tortillas and roll up tightly.

Step 8

Place burritos, seam side down, in pan. Cook over medium-high heat until sealed and light golden brown, about 2 minutes. Turn over and cook until light golden brown on other side.