Preheat the oven to about 375°F (190°C). Place poblanos over an open flame or under the broiler, turning until their outside is charred and blistered. Move them to a covered bowl or wrap in foil to steam; this loosens their skin for easier peeling.
Once the peppers are cool enough to handle, gently peel off the blackened skin. Slice along one side to open them, remove seeds and membranes. Fill each pepper with shredded Monterey Jack cheese, pressing gently but keeping peppers intact.
In a bowl whisk together eggs, flour, milk, baking powder, salt, and pepper until smooth. Place stuffed peppers in a greased baking dish. Pour the egg mixture over the peppers, letting it settle around them. Top with cheddar cheese.
Bake in the preheated oven for 30-35 minutes, watching for a puffed, golden top and bubbling cheese. Once done, allow the dish to rest for 5-10 minutes so it sets and slices come out nicely.