For the hard-boiled eggs: Fill a large, deep saucepan with a tight-fitting lid with 2 inches of cold water. Place the eggs in it and bring to a boil over medium-high heat. Immediately turn off the heat and cover with the lid. Let sit for 12 minutes. Transfer to a colander and rinse under cold water.
Once cooled, peel the eggs and slice in half lengthwise. Scoop out the yolks and place 4 yolks in each of 3 bowls. Mash well with a fork.
Line a large serving platter with the greens.
For the pickles and capers: Add the mayonnaise, chopped capers, mustard and gherkins to one of the bowls of yolks and mash together until thoroughly incorporated. Season with salt and pepper. Divide the filling among 8 egg halves. Top each with some capers and place on the platter.
For the wasabi and ginger: Add the mayonnaise and wasabi to the second bowl of yolks and mash together. Season with salt and pepper. Divide the filling among 8 egg halves. Top each with a slice of pickled ginger and a small piece of cucumber and place on the platter.
For the sesame carrot: Add the mayonnaise, sesame oil, sesame seeds and carrot to the remaining bowl of yolks and mash together. Season with salt and pepper. Divide the filling among the remaining 8 egg halves. Top each with more sesame seeds and place on the platter.
Adapted from "One Dish at a Time: Delicious Recipes from My Italian-American Childhood and Beyond" by Valerie Bertinelli © Rodale 2012. Provided courtesy of Valerie Bertinelli. All rights reserved.