If you have a preferred method of cooking a hard-boiled egg then feel free to use that, otherwise I suggest bringing a small pot of water to a boil, add the egg cold from the fridge then set a timer for 10 minutes. Reduce the heat a little after adding the egg so it's still boiling but not too strong, so the egg doesn't jump around and potentially crack. Turn now and then to try to center the yolk. Towards the end of the cooking time, prepare a bowl with iced water, then once the timer goes, transfer the egg to the iced water. Leave to cool a few minutes.
Meanwhile, chop the cornichons - how finely is up to you but I'd suggest at least as small as the capers, if not slightly smaller. You can roughly chop the capers or leave them, as you prefer. Chop the parsley (and tarragon, if using).
When ready, peel the egg then cut so that you can separate off the white and remove the yolk. Roughly chop the egg white and set aside.
Place the egg yolk in a small-medium bowl and add the vinegar and Dijon mustard. Mash or whisk well so that the yolk breaks up and doesn't have any lumps left, and everything is well combined.
Add the oil to the yolk mixture, a small amount at a time. After each addition, whisk the oil in so that it emulsifies. Make sure you have the oil largely all combined before you add the next drizzle. Keep adding a little at a time and whisking in until all the oil is added.
Add the chopped cornichons, capers, parsley and chopped egg white to the sauce and stir through so everything is well distributed. Use or store in a sealed container in the fridge for up to a day.