Put 1 cup of ice in each of 4 large, shallow, rimmed soup bowls, or small, decorative trays and press down gently to make a level bed. Sprinkle 1 tablespoon of the kosher salt over each bowl of ice. Place 6 clams in their shells on each bed.
In a small bowl, combine the white pepper, 1/8 teaspoon salt, the vinegar, olive oil, and shallots, and whisk to blend. Stir in the caviar.
Spoon an equal amount of the sauce over each, or place in small decorative ramekins and serve on the side.