Line a rimmed baking sheet with parchment paper or a silicone baking mat, making sure it lies completely flat with no wrinkles or bubbles.
Place candy canes in a heavy-duty plastic bag and seal tightly. Use a rolling pin to crush into small pieces, aiming for a mix of fine crumbs and slightly larger chunks.
Place chopped dark chocolate in a microwave-safe bowl and heat in 30-second intervals, stirring thoroughly between each interval until completely smooth. Stir in 1/2 teaspoon peppermint extract.
Pour melted dark chocolate onto the prepared baking sheet, spreading into an even layer about 1/4 inch thick. Tap the baking sheet gently on the counter to release air bubbles.
Place the baking sheet in the refrigerator and let the dark chocolate layer chill for about 15-20 minutes until firm to the touch but not completely hardened.
Place chopped white chocolate in a clean microwave-safe bowl and heat in 20-second intervals, stirring well between each heating until completely smooth. Stir in remaining 1/2 teaspoon peppermint extract.
Remove baking sheet from refrigerator and pour melted white chocolate over the set dark chocolate layer, spreading evenly to the edges and working relatively quickly.
Immediately sprinkle crushed candy canes evenly over the wet white chocolate layer. Press gently into the chocolate with your hands or the back of a spoon to help them adhere.
Return the baking sheet to the refrigerator and chill for at least 45 minutes to 1 hour, or until the bark is completely firm and set.
Remove fully set bark from refrigerator and lift it off the baking sheet using the parchment paper. Break into rustic, irregular pieces with your hands or cut into uniform squares with a sharp knife.