Chop the shallot into fine pieces and place it in a small pot, along with the white wine vinegar and one-third of the tarragon.
Heat the pot until it reaches a boil, then simmer on medium heat until the liquid has reduced by approximately 50%. Pour the contents through a strainer into a separate, clean bowl, retaining only the vinegar that's been infused with tarragon.
Cut the butter into small cubes and melt it in a skillet until it begins to bubble. Transfer the liquefied butter into a pitcher, removing any separated white solids.
In another container, mix together the egg yolks, 30ml of the chilled vinegar, and a pinch of sea salt to your liking. Employ a hand blender to blend the mixture for about half a minute until it starts to emulsify. Gradually add the melted butter in small amounts, continuing to blend as you incorporate it. If a hand blender isn't available, you can use a mini food processor or manually whisk the mixture in a bowl placed over a pot of gently boiling water.
Dice the rest of the tarragon finely, and incorporate it into the sauce along with the lemon juice. Keep the sauce in a warm location until you're ready to serve.