How to Make It
In a small bowl, toss the cucumber with 1/2 teaspoon of salt and let stand for 10 minutes. Rinse and drain well in a sieve. In the bowl, stir 1/4 cup of the cider vinegar with the sugar and 1/4 teaspoon of salt. Stir in the cucumber and refrigerate for 20 minutes. Drain.
Meanwhile, in a small saucepan, combine the remaining 1/3 cup of vinegar with the wine and shallot. Boil over moderately high heat until the liquid is almost completely evaporated, 6 to 8 minutes. Add the cream and simmer over moderately low heat for 5 minutes. Strain the mixture through a fine-mesh sieve into a bowl, pressing on the solids, then return to the saucepan; discard the solids. Off the heat, whisk in the butter, a couple of pieces at a time, adding more as the previous addition is incorporated. Stir in the lime juice and season with salt. Keep warm.
Arrange the oyster shells on a platter. In a small skillet, simmer the oysters with their liquor over moderate heat until the edges begin to curl, 1 to 2 minutes. Transfer each oyster to a shell. Spoon some of the beurre blanc on each and top with pickled cucumber and caviar. Serve immediately.