Preheat broiler. Lay chiles in a single layer on a baking sheet. Cook about 4 in. from broiler until chiles are blistering and black, about 5 minutes. Turn chiles over and broil until blistering and black all over, about 5 minutes. Put chiles in a large metal bowl and cover with foil or plastic wrap. Let sit 15 minutes.
Peel chiles and discard skins. Cut off stem ends and discard; remove seeds. Set chiles aside on layers of paper towels to dry.
Meanwhile, in a large frying pan over medium-high heat, cook chorizo, stirring occasionally to break up the meat, until cooked through, about 4 minutes.
Preheat oven to 375°. In a large bowl, mix chorizo, cotija, and oregano. Stuff chiles with mixture and lay them in an 8- by 12-in. baking dish.
In a large bowl, whisk eggs until they are thoroughly broken up and uniform in color and texture. Whisk in flour, baking powder, and salt. Sprinkle chiles with 1/2 the jack cheese. Pour egg mixture over chiles and sprinkle with remaining jack.
Bake until top starts to brown and the eggs are set but still soft, about 30 minutes.
Note: Nutritional analysis is per serving.