Add the eggs to a small pot with cold water to cover. Bring to boil, then lower the heat and simmer gently for 6 minutes. Remove the pot from the heat and use a slotted spoon to transfer the eggs to a bowl of ice water, cracking the eggs slightly to make them easier to peel. When cooled slightly, remove the shells with a spoon, then set the eggs aside to cool completely.
Meanwhile, combine the mayonnaise, garlic and Parmesan in a small bowl; mix well. Transfer to a pastry bag with the tip cut off.
Heat 2 tablespoons olive oil in a small sauté pan over medium heat. When the oil is glistening, add the anchovies and capers; sauté until beginning to crisp. Stir in the panko and cook until beginning to toast, about 3 minutes. Remove from the heat and set aside.
Cut the eggs in half lengthwise. Pipe the desired amount of mayonnaise mixture--1 to 2 tablespoons--onto each egg yolk. Top with the panko mixture, 1 shaving of Parmesan, and some lemon zest and parsley. Serve.