Preheat oven to 375°F. Lightly oil a 9x13-inch baking dish.
Arrange half the green chiles flat in the dish, opening them as needed.
Sprinkle half the Monterey Jack and half the cheddar over the chiles.
Layer remaining chiles on top, then add the rest of the cheeses.
Whisk eggs, milk, flour, baking powder, salt, and spices until smooth.
Pour the egg mixture evenly over the layered chiles and cheese.
Bake 35–40 minutes until puffed, golden, and the center is set.
Rest 10 minutes. Slice into squares and serve warm.