Line a 9 by 13 inch baking dish with aluminum foil, shiny side up. Smooth out any wrinkles. Heat 1 inch of water in a saucepan over low heat until steaming.

Put all but three quarters cup of the semi sweet chocolate in a heatproof bowl. Set the bowl over the saucepan of steaming water. Do not let the bowl touch the water. Stir until one third of the chocolate is melted. Remove the bowl from the saucepan. Keep the steaming water over low heat. Gradually stir the reserved 3/4 cup semisweet chocolate into the bowl, a few pieces at a time, until all the chocolate is melted.
Return the bowl to the saucepan, 5 to 10 seconds at a time, to help melt the chocolate, if needed. Do not rush this step. It may take up to 10 minutes to melt the chocolate.

Wipe off any moisture from the bottom of the bowl. Stir 3/4 teaspoon peppermint extract into the chocolate, then quickly pour into the prepared baking dish and spread in an even layer. Firmly tap against the counter to remove any air bubbles. Set aside at room temperature until almost set, about 10 minutes.

Meanwhile, put all but one cup of the white chocolate in a large bowl and repeat the melting process over the steaming water. Dry off the bottom of the bowl. Stir in the remaining 3/4 teaspoon peppermint extract. Pour over the semi sweet chocolate and spread in an even layer. Sprinkle immediately with the crushed candy canes, gently pressing into the white chocolate. Set aside at room temperature until firm, about one hour. Lift the bark out of the pan using the foil and break it into pieces. Store in an airtight container at room temperature for up to 2 weeks.