Prepare the Reduction: In a small saucepan, combine the white wine vinegar, white wine, shallot, and tarragon. Bring to a simmer over medium heat and cook until the mixture is reduced by half. This will take about 5-7 minutes. Remove from heat and let it cool slightly.
Prepare the Double Boiler: Fill a saucepan with a few inches of water and bring it to a simmer over low heat. Place a heatproof bowl over the saucepan, ensuring that the bottom of the bowl does not touch the water. This will be your double boiler.
Whisk the Egg Yolks: In the heatproof bowl, whisk the egg yolks until they are slightly thickened. This will take about 1-2 minutes.
Add the Reduction: Slowly pour the cooled reduction into the egg yolks, whisking continuously to combine.
Cook the Sauce: Place the bowl over the simmering water (double boiler) and continue to whisk the mixture. Slowly drizzle in the melted butter, whisking constantly, until the sauce thickens and becomes creamy. This will take about 5-7 minutes. Be careful not to overheat the sauce, as this can cause the eggs to curdle. If the sauce becomes too thick, you can add a little warm water to thin it out.
Season and Serve: Once the sauce has reached the desired consistency, remove it from the heat. Season with salt, pepper, and a squeeze of lemon juice, if desired. Serve the béarnaise sauce immediately over grilled steak, roasted vegetables, or eggs Benedict.
