Place the vinegar, water, shallot, peppercorns and tarragon stalks into a small saucepan over a medium heat and simmer, until you have half a tablespoon of liquid remaining. Strain the liquid into a bowl and set aside.
Place the egg yolk and tarragon vinegar reduction into a food processor and blend together until light and frothy.
With the food processor still running on its slowest speed, add the melted butter, 1 tbsp at a time, until the sauce is thick and smooth. Be careful not to over beat the mixture as it may separate.
Stir in the chopped tarragon and season, to taste, with salt and freshly ground black pepper.