Meanwhile, make your sauce. Hard boil the eggs by placing them in a pot with enough water to cover. Bring to a boil, let boil for a minute. Then turn off the heat and cover. Let them sit for 9 minutes.
Drain the eggs and run them under cold water. As soon as they are cool enough to handle, peel the eggs.
Chop the hard boiled eggs finely and add them to a mixing bowl. Add the chopped anchovy and minced parsley.
Right before you're ready to serve, melt the butter gently, then pour it into the mixing bowl with the other sauce ingredients. Drizzle with a few drops of freshly squeezed lemon. Mix everything together briefly and gingerly, taking care not to mash the bits of egg yolk. Taste and, if needed, season with salt.