New Mexican–Style Breakfast Burritos

Name: New Mexican–Style Breakfast Burritos
Ingredients: 10–12 dried New Mexico chiles (about 2 oz.), seeds removed; 3 garlic cloves, smashed; ¼ tsp. Mexican oregano; ¾ tsp. Diamond Crystal or ½ tsp. Morton kosher salt; 1 Tbsp. plus 1½ tsp. all-purpose flour; 8 oz. fresh chorizo, breakfast sausage, or plant-based sausage; 1 large russet potato (about 10 oz.); ¾ tsp. garlic powder; ¾ tsp. Diamond Crystal or ½ tsp. Morton kosher salt, plus more; ¼ tsp. freshly ground black pepper, plus more; 3 Tbsp. extra-virgin olive oil, plus more; 4 large eggs; 1 Tbsp. chilled unsalted butter; 2 9–10 flour tortillas; 1 cup shredded cheddar; ¼ cup jarred New Mexico green chiles