In a small saucepan, combine the white wine vinegar, white wine, shallots, and 1 tablespoon of the chopped tarragon. Bring the mixture to a boil over medium heat, then reduce the heat and simmer until the liquid has reduced by half, about 5-7 minutes.

In a small heatproof bowl, whisk the egg yolks until they are light and frothy.

Add 1 tablespoon of the reduced liquid from the saucepan to the egg yolks and whisk vigorously to combine. Repeat this process with another tablespoon of the reduced liquid.

Return the saucepan with the remaining reduced liquid to the stove over low heat. While whisking the egg yolks constantly, slowly pour the egg mixture into the saucepan in a thin stream.

Continue to whisk the mixture until it thickens and doubles in volume, about 3-5 minutes.

Remove the saucepan from the heat and slowly pour in the melted butter, whisking constantly until the sauce is smooth and creamy.

Add the remaining chopped tarragon to the sauce and season with salt and freshly ground black pepper to taste.

Serve the Bearnaise sauce immediately with steak, grilled fish, or vegetables.