Preheat broiler. Lay chiles in a single layer on a baking sheet. Cook about 4 inches from broiler until blistering and black, about 5 minutes. Turn chiles over and broil until blistering and black all over, about 5 minutes. Put chiles in a plastic bag and let sit 15 minutes.
Peel chiles and cut out stem and remove seeds. Don't worry if they tear. Set on paper towels.
Meanwhile, cook chorizo, stirring occasionally to break up the meat, until cooked through. Drain on paper towels.
Preheat oven to 350 degrees. In a large bowl, mix chorizo, cotija, and oregano. Stuff chiles with mixture and lay them in an 8 by 12-inch baking dish.
In a large bowl, whisk eggs. Whisk in flour, baking powder, salt, and cottage cheese. Sprinkle chiles with half of cheddar or jack. Pour egg mixture over chiles and sprinkle with remaining cheese.
Bake until top starts to brown and the eggs are set but still soft, about 45 minutes. Serves 6 to 8.