Combine the white pepper, salt, wine, olive oil, and shallots in a small bowl and whisk to blend. Set aside.
Put 1 cup of the ice in each of eight small shallow rimmed soup bowls and press down gently to make a level bed. Sprinkle 1 tablespoon of the kosher salt over each bowl of ice. Place 3 oysters in their shells on each bed and spoon an equal amount of the sauce over each. Dot with the caviar and serve cold.
Makes 8 servings.