Line a 9x13-inch rimmed baking sheet with parchment paper, allowing the edges to overhang for easy lifting.

Melt the dark chocolate in a heatproof bowl set over a pot of barely simmering water (double boiler) or in the microwave in 30-second bursts, stirring until silky smooth. Stir in 1/2 teaspoon of peppermint extract.

Pour the melted dark chocolate onto the prepared baking sheet and spread into an even 1/4-inch rectangle using an offset spatula. Tap the pan against the counter to release air bubbles. Chill in the refrigerator for only 20 to 25 minutes until just barely set to the touch (do not let it become ice cold).

While the dark layer chills, melt the white chocolate using the same gentle double boiler method. Stir in the remaining 1/4 teaspoon peppermint extract. Let the melted white chocolate cool for 5 minutes so it is lukewarm, not hot.

Gently pour the lukewarm white chocolate over the chilled dark chocolate layer, carefully smoothing with an offset spatula. Work quickly and gently to avoid melting the bottom layer.

Immediately scatter the crushed candy cane pieces across the wet white chocolate, pressing down very gently with your fingertips to adhere. Chill in the refrigerator for 30 to 45 minutes until completely firm. Lift out using the parchment overhang and break into rustic shards!