Combine the tarragon, shallots, 1/4 teaspoon of the salt, pepper, and vinegar in a small saucepan. Simmer the ingredients over medium-high heat until almost all the vinegar evaporates and 1 tablespoon is left, about 15 minutes. Remove the pan from the heat and allow the syrup to cool slightly.
In the top of double boiler, or in a stainless steel bowl set over a pot of simmering water, whisk the egg yolks and cayenne into the syrup over a very low heat. As soon as the egg yolks have thickened, slowly add the melted butter a bit at a time, whisking continuously. Add the lemon juice and adjust the seasoning to taste, then whisk in the fresh herbs. Serve immediately over the Chicken Clemenceau, or keep warm, covered, over a pot of simmering water, for a short time.
Yield: 1 cup