Place the shallot, vinegar, and wine with salt and pepper in a small saucepan and reduce it until it’s almost dry. Add the water and remove from the heat to cool somewhat.
Strain it if you wish to remove the shallot.
Place a good-sized stainless steel bowl over another pot with simmering water (a.k.a. a bain-marie). The water in the pot should be low enough so that it does not touch the bottom of the bowl.
Combine the vinegar, wine, and water mixture and the egg yolks in the bowl.
Begin whisking incessantly until the yolks form ribbons, lighten in color, and increase in volume. Ensure the heat under the simmering water is not too high but just enough to keep it lightly simmering. Too much heat, and the eggs can begin to scramble.
You can even briefly remove the bowl from the heat from time to time to limit the temperature.
Add the butter in increments, constantly whisking. Once the butter is fully incorporated, strain the sauce to remove any coagulated egg particles.
Finish with the tarragon and additional salt and pepper if necessary.
