In a saucepan, bring the vinegar and wine to a boil with the shallots and tarragon sprig. Season with pepper. Reduce until only about 45 ml (3 tablespoons) of liquid remains. Strain.
In a bowl over a saucepan of simmering water, with a whisk, beat the vinegar reduction and egg yolks until thick and foamy.
Remove from the double boiler. Off the heat, drizzle in the butter, whisking constantly. Add the chopped tarragon. Adjust the seasoning. If the sauce seems too thick, whisk in a little hot water. Keep at room temperature and just before serving, place back on the double boiler, whisking for just a few seconds, to warm it up.