In the bowl of a food processor, puree the avocados with the lemon juice. Add the shallots, garlic, salt, and white pepper, and pulse to combine, scraping down the sides to incorporate all the ingredients. Transfer to a large bowl.

In a medium bowl, beat the cream with an electric mixer until stiff. Fold the cream into the avocado puree and adjust the seasoning, to taste. Transfer to a pastry bag fitted with a star tip.

Arrange the oysters on trays of ice that have been lightly sprinkled with kosher salt. Pipe a heaping tablespoon of the mousse onto each oyster. Top with a dollop of caviar and sprinkle with chives. Garnish the trays with lemon wedges and serve with cocktail forks.