Line a 12x18-inch baking sheet with parchment paper.
In a microwave-safe bowl, melt the milk chocolate in 30-second intervals, stirring after each melting, for a total of 1 to 5 minutes (depending on your microwave), until melted and smooth.
Stir in the peppermint extract. Spread the chocolate evenly in the prepared pan. Place the pan of chocolate in the refrigerator to chill until mostly set, about 15-20 minutes.
Meanwhile, melt the white chocolate in a microwave-safe glass or ceramic bowl in 30-second intervals, stirring after each melting, for 1 to 5 minutes. Stir in ¼ cup of the crushed candy canes.
Spread the white chocolate mixture evenly over the chilled and mostly set milk chocolate. Sprinkle the remaining candy cane pieces evenly over the white chocolate layer. Chill in the refrigerator until set, about 1 hour.
When set, break into small pieces to serve.