Preheat your oven to 375°F (190°C). Grease a 9×13 inch baking dish with cooking spray or butter.
Roast the poblano chiles directly over a gas flame or under the broiler until the skins are completely charred and blistered. Place the charred chiles in a bowl and cover tightly with plastic wrap or a kitchen towel. Let them steam for about 10 minutes. Then, remove the charred skin by rubbing it off with your fingers or using a paper towel. Make a slit down the side of each chile and remove the seeds and membranes.
In a medium bowl, whisk together the eggs, milk, flour, baking powder, salt, black pepper, and cayenne pepper until smooth.
Spread half of the shredded Monterey Jack cheese in the bottom of the prepared baking dish. Place the prepared poblano chiles over the cheese in a single layer. Sprinkle the remaining Monterey Jack cheese over the chiles, followed by the shredded cheddar cheese.
Pour the egg mixture evenly over the top of the cheeses and chiles.
Bake the casserole in the preheated oven for 35-40 minutes, or until the top is golden brown and the center is set