Place butter in a small saucepan over medium-low heat; cook 5 minutes or until completely melted. Carefully skim solids off the top with a spoon; discard solids. Slowly pour remaining butter out of pan, leaving remaining solids in pan; discard solids.
Combine egg yolks and 2 tablespoons water in a small saucepan, stirring with a whisk until foamy. Place pan over medium heat, stirring constantly until mixture thickens slightly. Gradually add 1/4 cup clarified butter, about 1 tablespoon at a time, stirring with a whisk until each addition is incorporated and mixture is thick. Reserve remaining clarified butter for another use.
Combine wine, vinegar, shallots, and tarragon sprigs in a heavy saucepan; bring to a simmer. Cook until reduced to 2 tablespoons. Strain through a sieve into butter mixture. Stir in salt, pepper, and chopped fresh tarragon.