Preheat the oven broiler with the rack about 6 inches from the heat source. Cut the tips from the asparagus and set them aside. Cut the remaining part of the spear into 2-inch chunks and set aside. Mix the eggs, milk and parsley together in a bowl and set aside. Heat the olive oil and butter in a large sauté pan over medium heat. When the butter has melted and looks foamy, add the onion and cut asparagus spears and cook for about 2 minutes or until softened. Add the asparagus tips and cook for another minute or until the vegetables are crisp-tender. Scatter the cheeses over the vegetables. Pour in the egg mixture and turn the heat to low. Stir once or twice, then cook undisturbed for about 8 minutes or until the bottom has set. Place the pan under the broiler for up to a minute or until the frittata is puffed and golden. Season to taste with freshly ground black pepper.

To bake the frittata, preheat the oven to 375 degrees. After the frittata bottom has set on the stovetop, bake it for about 12–15 minutes, until the eggs are set and are puffed and golden brown.