In a small saucepan over medium-low heat, combine
the shallot, most of the tarragon, a sprinkle of salt
and pepper, and the vinegar. Cook until all but about 2
tablespoons of the vinegar has evaporated, just a minute
or two. Let cool.
Beat the egg yolks with 1 tablespoon water and stir
into the vinegar mixture. Return to the stove over low
heat and beat continuously with a wire whisk until thick,
about 5 minutes.
With the heat as low as possible, use a wooden
spoon to stir in the butter 1/2 teaspoon or so at a time.
Add the remaining tarragon and taste; add salt and pepper
if necessary and, if the taste is not quite sharp
enough, a bit of lemon juice. If the sauce is too thick, stir
in hot water, a teaspoon at a time. Serve immediately.