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 half of the bittersweet chocolate in a heatproof bowl. Set the bowl over the saucepan of steaming water (do not let the bowl touch the water) and heat, stirring, until most of the chocolate is melted. Remove the bowl from the saucepan; keep the steaming water over low heat. Gradually stir in the remaining bittersweet chocolate a few pieces at a time, until all of the chocolate is melted. Return the bowl to the saucepan if needed, 5 to 10 seconds at a time, to help melt the chocolate. 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 of the peppermint extract into the chocolate, then quickly pour into the prepared baking dish and spread in an even layer. Firmly tap the bottom of the dish against the counter to remove any air bubbles. Set aside.
Meanwhile, put all 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 1/4 teaspoon peppermint extract.
Dollop tablespoons of the white chocolate mixture over the bittersweet chocolate. Drag a toothpick in loops and swirls through the white and dark chocolate to create a marbling effect. Sprinkle immediately with the chopped peppermint patties and crushed candy canes, gently pressing them into the chocolate. Set aside at room temperature until firm, about 1 hour.
Lift the bark out of the dish using the foil and break it into pieces. Store in an airtight container at room temperature for up to 2 weeks.
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