Bake piecrust until partially cooked.
Soften the onion in the oil, covered with a lid, for 5 minutes. Steam the broccoli florets until tender but still slightly crisp, 5-6 minutes. Mix them with the onion and season with salt and pepper. Spread over bottom of the pastry shell and intersperse with slices of cheese.
Beat the eggs thoroughly and stir in the milk or cream plus the extra milk. Season to taste with salt and pepper. Pour the mixture over the broccoli. Bake in a preheated 400 oven until lightly browned and set, about 30 minutes. Serve hot or warm.