Clarify the butter by melting it in a small, heavy-based saucepan over a low heat. When the butter is foaming, remove the pan from the heat and leave it to stand for a few minutes so that the white solids sink to the bottom of the pan. Sieve the butter through a fine sieve and discard the solids.
Pour the vinegar into a non-reactive saucepan. Add the shallots, chopped tarragon and salt, to taste. Heat gently over a medium heat until the volume of liquid has reduced by more than half. Strain and set aside until completely cooled.
Lightly beat egg yolks with one teaspoon of water. Stir the egg yolk mixture into the cooled vinegar, then add the lemon juice.
Pour the mixture into a bowl suspended over a pan of simmering water (do not allow base of the bowl to touch the water). Whisk constantly until the sauce has thickened enough to coat the back of a spoon and has increased in volume.
Remove the bowl from the heat and slowly pour in the clarified butter in a steady stream, whisking continuously, until the mixture is thick and smooth. Fold in the tarragon leaves and season, to taste, with salt and freshly ground black pepper.