In a small saucepan mix shallots, black pepper, white wine, red wine vinegar, and tarragon stems. Reduce the mixture until most liquid is evaporated. Transfer into a small pan and reserve in the refrigerator.
Prepare a Bain Marie by bringing water to a boil in a large saucepan. Once boiling, reduce heat under the Bain Marie to maintain a slow simmer. Put egg yolks and water in small salad bowl and whisk them while holding the bowl just above the simmering water in the Bain Marie. Beat the yolks until they start to thicken (6-7 minutes). Add butter slowly (stirring in 2 ounces at a time) until sauce achieves a creamy consistency. Add 1 T of Bearnaise reduction. Add salt and pepper to taste. Finish with 1 T thinly chopped tarragon.