Fill a large pot with a tight fitting lid about ¼ full with cool water - Place eggs in a single layer on the bottom of the pot - Add enough cool water to cover eggs by 2-3 inches - Place over high heat - Bring to a rolling boil
Once a boil has been achieved, turn off the heat, cover, and set aside for 10-12 minutes - After 10-12 minutes, transfer the eggs to the ice bath and allow to cool for apx 5 minutes
Once the eggs have cooled, peel and cut in half - Remove the yolks and transfer them to a medium bowl - Set aside until needed
Place the anchovies, capers, and parsley in a mortar and pestle and mash into a smooth paste - Add the mayonnaise, olive oil, and lemon juice (transfer the anchovy/caper paste to a medium bowl if necessary) - Add the tuna (if using) - Mix until thoroughly combined - Season with salt and pepper to taste
Mash the egg yolks with a fork until smooth - Transfer to the filling mixture and thoroughly combine
Use a teaspoon or piping bag to transfer the filling mixture evenly into the hollows of the egg whites (you want to mound the filling a bit)
Garnish each with a slight sprinkle of finely chopped parsley and a couple of capers (if desired)