Chill a large mixing bowl and beaters for 10 minutes to help the cream whip quickly.
Whip the cold heavy cream to medium peaks; do not overbeat. Set aside.
Beat cream cheese, confectioners' sugar, and vanilla until smooth and fluffy.
Fold the whipped cream into the cream cheese mixture until just combined and airy.
Spread a thin swipe of caramel in a 9-inch square dish to keep the first layer in place.
Arrange a snug layer of crackers in the dish. Spread about one-third of the cream over the crackers.
Drizzle a few spoonfuls of caramel over the cream and sprinkle a pinch of flaky sea salt.
Repeat the layers—crackers, cream, caramel, salt—until you have 3 to 4 layers, ending with cream and a caramel drizzle.
Cover tightly and refrigerate at least 6 hours, preferably overnight, until the crackers soften into cake-like layers.
Finish with chocolate shavings and a final pinch of flaky salt. Slice and serve cold.