Preheat the oven to 350 degrees F (175 degrees C). Lightly coat a 9x9-inch baking dish with cooking spray.

Arrange pizza squares in prepared baking dish.

Whisk eggs, milk, mozzarella, onion, butter, garlic, oregano, and salt together in a large bowl; pour over pizza squares. Top with Parmesan and pepper flakes. Cover dish with aluminum foil.

Bake in the preheated oven 45 minutes. Remove foil and continue baking until eggs set and Parmesan melted, about 20 minutes more.