Preheat the oven to 400 degrees F (200 degrees C). Grease a 9x13-inch baking dish.

Split chiles open and spread half of the chiles into the bottom of the prepared baking dish. Layer with Cheddar cheese and top with remaining canned chiles. Whisk eggs, flour, and evaporated milk together in a bowl; pour mixture over chiles and cheese.

Bake in the preheated oven until set, about 30 minutes; remove casserole, and top with Monterey Jack cheese. Pour green enchilada sauce over the top; return to the oven and bake until cheese is melted, about 15 more minutes.