Prep the pan: Line the bottom of a 9-inch springform pan with parchment.
Lightly grease the sides to help with a clean release.
Make the crust: Pulse the cookies (filling and all) into fine crumbs. Stir in melted butter until the mixture feels like damp sand.
Press and chill: Firmly press crumbs into an even layer on the bottom and about 1 inch up the sides. Use a flat-bottom glass to compact.
Chill while you make the filling.
Melt the chocolates: Place dark, milk, and white chocolate in three separate heatproof bowls. Microwave each in 20–30 second bursts, stirring between, until smooth. Let cool to just warm, not hot.
Whip the cream: In a cold bowl, whip heavy cream to stiff peaks.
Beat in 1/4 cup powdered sugar. Set aside in the fridge.
Beat the cream cheese: In a large bowl, beat cream cheese until very smooth, 2–3 minutes. Add remaining 1/2 cup powdered sugar, vanilla, and fine salt (plus espresso powder if using).
Beat until fluffy and no lumps remain.
Lighten the base: Gently fold half of the whipped cream into the cream cheese until combined. Fold in the rest. Don’t deflate it.
Divide and flavor: Split the cheesecake base evenly into three bowls.
Stir dark chocolate into one, milk chocolate into another, and white chocolate into the last. Work quickly so the chocolate doesn’t seize.
Layer one: Spread the dark chocolate mixture over the crust. Smooth the top with an offset spatula.
Chill 10 minutes to lightly set.
Layer two: Add the milk chocolate layer. Smooth gently so you don’t disturb the base. Chill another 10 minutes.
Layer three: Top with the white chocolate layer.
Smooth the surface. Tap the pan lightly on the counter to release air bubbles.
Chill to set: Cover and refrigerate at least 6 hours, preferably overnight. The longer chill helps clean slices.
Finish and serve: Run a thin knife around the inside edge, then release the springform ring.
Garnish with chocolate curls, a dusting of cocoa, or a pinch of flaky salt. Slice with a hot, dry knife for neat edges.