Bring a large saucepan of water to a boil over high heat. Gently place the eggs in the boiling water (use a spoon or ladle, and work quickly but carefully) and add water to cover by at least an inch. Allow the water to boil for 30 seconds, then remove the pot from the heat and let the eggs sit in the water for 9 minutes. Drain and rinse under cool water for a few minutes. Refill the pot with cold water and return the eggs to the pot of water. Tap the warm eggs lightly on the counter, and give them a quick roll to crackle up the shells, and then peel carefully while they are submerged in water. Cool the peeled eggs, then chop them as roughly or as finely as you like, using a knife or a food processor.
While the eggs are cooling, in a medium bowl, combine the mayonnaise, lemon zest and juice, parsley, dill, shallot, capers (if using), and salt and pepper until well blended. Add teh cooled chopped eggs and gently fold them in. Check and adjust seasonings as needed.