Line a 9-inch square baking pan with parchment, leaving overhang on two sides for easy lifting. Preheat the oven to 325°F (165°C).
Crust: In a bowl, combine the cookie crumbs, melted butter, and salt until the mixture resembles wet sand. Press firmly into the prepared pan, packing it tightly with the bottom of a measuring cup. Bake for 8 minutes, then cool to room temperature (about 15 minutes).
White chocolate base: Place the chopped white chocolate and 1/3 cup cream in a heatproof bowl set over a saucepan of barely simmering water (do not let the bowl touch the water). Stir until completely smooth and glossy, 3–5 minutes. Remove from heat and cool until just lukewarm and slightly thickened, about 10 minutes.
Creamy filling: Beat the cream cheese and granulated sugar on medium-high until fluffy and smooth, 2 minutes, scraping the bowl as needed. Mix in the peppermint and vanilla extracts just until combined.
Blend in the cooled white chocolate mixture on low speed until silky and fully incorporated; do not overbeat.
Whipped cream: In a chilled bowl, whip 1 cup cold cream with the powdered sugar to stiff peaks, 2–3 minutes. Fold half of the whipped cream into the cheesecake base to lighten it, then gently fold in the remaining half just until no white streaks remain. Fold in 3/4 cup of the chopped peppermint bark, reserving the rest for topping.
Assemble and chill: Spoon the filling over the cooled crust, smooth the top, and tap the pan a few times on the counter to release air bubbles. Cover and refrigerate until fully set, at least 6 hours or overnight for the cleanest slices.
Finish and serve: Sprinkle the reserved peppermint bark and white chocolate curls over the set cheesecake. Lift out using the parchment sling and cut into 16 bars with a hot, dry knife, wiping the blade between cuts. Keep chilled until serving.