In a saucepan, place the eggs and cover with cold water. Bring to a boil over medium heat and cook for 5 minutes, beginning your time when the water just begins to boil.
Drain and place the eggs into cold water. Gently roll each egg between your thumb and forefinger to break the shell, then remove from the thickest end towards the thinnest end of the egg. Set the shelled eggs aside in water until ready to serve.
In a bowl, whisk together the mustard, vinegar and egg yolk. Pour the oil in a thin stream, whisking constantly. Add the tarragon. Season with salt and pepper.
Pour the mayonnaise in a pretty ramekin and place the eggs in egg cups. Serve with toothpicks with which each guest can dip the egg in the mayonnaise.