In a small saucepan, combine the white wine, vinegar, shallot and tarragon. Cook over medium heat for 8 to 10 minutes until the liquid in the pan is reduced to 2 tablespoons. Remove from the heat and set aside to infuse and cool for about 5 minutes.
Strain the liquid through a very fine mesh strainer, pushing on the solids to extract as much liquid as possible. (You should have about 2 tablespoons).
Meanwhile, place the butter in a liquid measuring cup. Cover and microwave on high for 75 to 90 seconds, until the butter is melted and the white solids start to sink to the bottom. Set it aside for 5 minutes, until the solids settle and separate from the golden, clarified butter. (You can also melt the butter in a small pot on the stove to make this clarified butter).
In a tall, 2-cup liquid measuring cup, combine the egg yolks, vinegar reduction, and salt. Using a stick blender, blend the mixture until it’s very smooth.
With the blender on low speed, stream in the warm, clarified butter from the top of the liquid measuring cup (leave the milk solids in the bottom of the cup, and discard.) You should be adding around ¾ cup of clarified butter. Start by adding a few drops of butter at a time, and then increase speed to add a slow stream of butter as the mixture starts to emulsify and thicken.
Stir in the lemon juice, pepper, and up to 1 teaspoon more of chopped fresh tarragon. Adjust the consistency with warm water, by stirring in 1 teaspoon at a time. (You should be able to drizzle the sauce so that it coats a steak or vegetables. It should not be thick like mayonnaise.) Serve immediately.
Notes
Separate the eggs when they’re cold. It is much easier to separate the whites and yolks when the eggs are cold. Then, leave the yolks out for about 15 minutes to give them time to come closer to room temperature. This will make them easier to blend and reduce the risk of the cold eggs solidifying the melted butter.
If you over-reduce the wine and vinegar mixture, it’s okay. Just add enough water to reach 2 tablespoons of liquid after it’s strained.
For the clarified butter, you can melt the butter on the stove in a small pot instead of using the microwave.
Use ghee. Ghee is clarified butter, so if you have it, you can use ¾ cup of melted ghee instead of clarifying your own butter.
It’s best to add the butter when it’s hot, around 140°F to 150°F. The hot butter will cook and thicken the raw eggs as the mixture emulsifies.
Add chervil: A classic French bearnaise sauce is finished with freshly chopped chervil instead of tarragon (tarragon is easier to find, though!). Use it if you have it, or you can add ½ teaspoon of freshly chopped parsley for a bright, grassy flavor.