Line a 9x13-inch baking pan with parchment paper, letting the paper hang over the edges for easy removal later.
In a microwave-safe bowl, melt 2 cups of the semisweet chocolate chips in 30-second increments, stirring between each one, until completely smooth. This usually takes about 2 minutes total.
Stir in 1/2 teaspoon of peppermint extract until well combined.
Pour the chocolate into the prepared pan and spread it into an even layer. Let it sit at room temperature for 10 minutes while you prepare the white chocolate. The chocolate should look matte on top but still be slightly soft if you gently touch it.
In a clean microwave-safe bowl, melt the white chocolate chips in 30-second increments, stirring between each one, until smooth. Be careful not to overheat white chocolate as it can scorch easily.
Stir in the remaining 1/2 teaspoon of peppermint extract.
Pour the white chocolate over the semisweet chocolate layer. Use a spatula to spread it gently into an even layer, being careful not to disturb the layer underneath.
Melt the remaining 1/2 cup of semisweet chocolate chips in a microwave-safe bowl. Use a spoon to add small dollops of the melted chocolate randomly over the white chocolate layer.
Use a butter knife or offset spatula to swirl the semisweet chocolate into the white chocolate, creating a marbled pattern. Don't over-swirl or the colors will muddy together.
Immediately sprinkle the crushed candy canes over the top, pressing them down gently so they stick to the chocolate.
Refrigerate for at least 30 minutes, or until the chocolate is completely firm and set.
Once set, lift the bark out of the pan using the parchment paper. Place it on a cutting board and break it into irregular pieces by hand, or cut it into squares with a sharp knife.
Store in an airtight container in the refrigerator for up to 2 weeks.