Heat a tablespoon of the butter in a wide heavy-based saucepan over medium heat. Add the shallot and sweat for a couple of minutes, without allowing it to brown.
Deglaze the pan with the champagne vinegar. Simmer gently until the liquid has almost evaporated.
Add the wine and fish stock and reduce until 3 tablespoons remain.
Cut the remaining butter into large cubes.
Begin swirling the cold butter into the liquid over medium heat. The liquid should be hot and it doesn't matter if it boils. Stir constantly and continue adding the butter until the desired consistency is achieved. Wait until each piece of butter has emulsified before adding the next. The sauce should coat the back of a spoon.
Season to taste just before serving and stir in 1 teaspoon of black caviar per person.