Preheat the oven to 300 degrees. Grease a 9-inch round cake pan or 8 x 8-inch baking dish and line it with unbleached parchment paper. I personally prefer a springform pan or a removable bottom tart pan.
In the bowl of a stand mixer fitted with the paddle attachment (or use a hand mixer), beat the butter, sugar and vanilla until combined, about 1 minute. Stir in the flour and salt and beat until the mixture comes together into a ball of dough.
Press the mixture evenly into the prepared pan. Bake until the surface just starts to turn golden, about 25-30 minutes. Cut into wedges while still warm and allow to cool completely. Separate the wedges a bit before drizzling with chocolate.
In a heat-proof bowl set over a saucepan of simmering water (or use a double boiler), melt the semisweet chocolate and 1 teaspoon coconut oil together until smooth. Stir in ½ teaspoon peppermint extract. Spread over the cooled shortbread. If you want to spread melted white chocolate over the semisweet chocolate, freeze until set, about 5 minutes. If you plan to drizzle the melted white chocolate, you can just go straight to that step.
Melt the white chocolate, 1 teaspoon coconut oil, and ½ teaspoon peppermint extract in the same way. Spread or drizzle over the semisweet chocolate layer. Immediately sprinkle with crushed candy canes or freeze-dried fruit. Freeze until firm, about 5-10 minutes and store in the refrigerator.