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 3/4 cup of the semisweet chocolate in a heatproof bowl. Set the bowl over the saucepan of steaming water (do not let the bowl touch the water) and 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 of 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 the dish against the counter to remove any air bubbles. Set aside at room temperature until almost set, about 10 minutes.

Meanwhile, put all but 1 cup of the white chocolate in a large heatproof 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 semisweet chocolate and spread in an even layer. Sprinkle immediately with the crushed candy canes, gently pressing them into the white chocolate. Set aside at room temperature until firm, about 1 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.