1. Stack tortillas and cut into about 1/2-inch squares.
2. In an 11- to 12-inch frying pan, combine 1 teaspoon oil and the onion. Stir often over medium-high heat until onion is limp, about 3 minutes. Add tortillas and remaining oil, turn heat to high, and stir until small bits start to brown, 8 to 10 minutes.
3. Scatter tortilla mixture in an 8- by 12-inch oval or rectangular casserole (at least 2 in. deep).
4. Shred 2 cups cheese; sprinkle 1 cup over the tortilla mixture. Cut remaining cheese into sticks about 1/4 inch thick.
5. Drain chilies and stuff equally with cheese sticks. Lay chilies in a single layer in casserole. Sprinkle with cilantro.
6. Separate eggs. Whip whites with a mixer on high speed until they hold soft but distinct peaks; set aside.
7. Mix egg yolks (use unwashed beaters) with flour, then beat on high speed until paler in color, about 2 minutes. Beat half of the whites into the yolks, then fold in remaining whites and pour mixture over chilies, scraping bowl. Top egg mixture with shredded cheese.
8. Bake in a 375° oven until topping is golden brown and feels firm when lightly touched, about 15 minutes. Cutting through chilies, scoop from casserole to serve.