Start clarifying the butter, if you haven't already but make sure you cool the butter off before using it! It should be warm but not hot - just hot enough to melt and turn clear but obviously less than 100°C.

In a small pot, combine the vinegar, shallots, cracked pepper and tarragon leaves. Bring to a boil and reduce until nearly dry. Remove from the heat and set aside. Place the egg yolks in a warm metal mixing bowl though glass worked perfectly well for me Add the tarragon reduction (and, if you're feeling cowardly and unambitious, a few drops of water to give a little insurance against curdling). Bring a few cups of water to a boil in the bottom half of the double boiler and place the metal bowl with the eggs over the simmering water. Whisk the eggs constantly, pulling the bowl away from the heat if the eggs start to curdle or harden.

Once the eggs have become very foamy and start to thicken, slowly begin to incorporate the clarified butter, whisking as you add it. (A tea towel "collar" can be useful to keep the bowl in place whilst you whisk.) Keep whisking. You're making emulsion here. Whisk until all the butter is incorporated. Season with salt and pepper and mix in the fresh choped tarragon sprig.
Serve immediately, or hold in a thermos (but not for longer than an hour).