In a small saucepan over medium-high heat, combine the vinegar, shallot and the 2 tsp tarragon. Bring to a boil, reduce the heat to low and simmer until the vinegar is almost completely evaporated, 3 to 5 minutes. Remove from the heat and let cool slightly, then strain into a heatproof bowl. Add the water and egg yolks. Set the bowl over but not touching simmering water in a small saucepan and whisk until thick and pale yellow, about 3 minutes.
Transfer the mixture to a blender and blend until smooth. Let cool for 1 minute. With the motor running, slowly pour in the clarified butter and blend until incorporated. Add the 1/2 tsp lemon juice, or to taste, and the 1 Tbsp tarragon. Season with salt and white pepper and process to combine. Transfer the sauce to a double boiler and keep warm over very low heat until ready to serve.
One of the classic French sauces, béarnaise is traditionally served with beef. It can also be drizzled over grilled fish, such as salmon, or paired with vegetables or egg dishes.