Line a 12×18 inch jelly roll pan with aluminum foil.

Melt the milk chocolate in a microwave-safe glass or ceramic bowl in 30-second intervals, stirring after each melting, for 1 to 5 minutes (depending on your microwave). Do not overheat or chocolate will scorch. Stir in the peppermint extract. Spread the chocolate evenly in the prepared pan; chill until set, about 30 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 1/4 cup of the crushed candy canes. Spread the white chocolate mixture evenly over the milk chocolate. Sprinkle the remaining candy cane pieces evenly over the white chocolate layer. Chill until set, about 1 hour. Break into small pieces to serve.