Preheat oven to 350°F. Pulse crackers in a food processor until finely ground, 6 to 8 times. Add egg white, butter, and 3 tablespoons of the sugar. Pulse until well combined, about 4 times.

Scatter crumb mixture evenly on bottom of a 9- or 10-inch tart pan with removable bottom. Using a straight-sided glass or dry measuring cup, firmly press crumbs on bottom and against sides of pan.

Bake crust in preheated oven until golden brown around the edges, 12 to 14 minutes. Cool completely on a wire rack, about 30 minutes.

Meanwhile, pour cream into a small saucepan. Using a small knife, split the vanilla bean lengthwise. Using back of knife, scrape seeds from bean into cream; discard pod. Sprinkle gelatin evenly over cream. Let stand 5 minutes. Stir in remaining ½ cup sugar. Cook over medium-low, stirring constantly, until gelatin and sugar are dissolved, 4 to 5 minutes.

Transfer cream mixture to food processor. Pinch cream cheese into 1- to 2-inch pieces, and place in food processor. Add buttermilk and salt. Process mixture until smooth, 20 to 30 seconds; pour into cooled tart crust. Chill until set, 8 hours or up to 1 day. Before serving, scatter berries over top; garnish with basil leaves.