Begin by melting the chocolate. Place it in a heat-safe bowl and set it over a pan of gently simmering water to create a double boiler; stir until the chocolate is completely smooth and melted. Alternatively, melt the chocolate in the microwave, heating it in 30-second intervals and mixing thoroughly between each interval. Once the chocolate is fully melted, remove the bowl from the heat and set it aside to cool the chocolate slightly, about 10 minutes.
In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using a handheld mixer, beat the butter on high speed until it’s completely smooth and creamy, 2 to 3 minutes. Switch to medium-high speed and beat in the powdered sugar, cream, vanilla, peppermint extract, and salt. Once combined, switch to low speed and slowly pour the slightly cooled white chocolate in, with the mixer running. Beat until the ingredients are smooth and fully combined.
Using a large, serrated knife, slice a thin layer off the tops of the cakes to create a flat surface, if needed. Place one cake layer on your cake stand or serving plate, and evenly cover the top with a large dollop of frosting. Top with the second layer, placing it upside down (so the flat bottom of the cake is on top), and spread a thin layer of the frosting all over the top and sides for your crumb coat. Smooth the frosting with an offset spatula and bench scraper and chill the cake until the frosting has hardened slightly,10 to 20 minutes.
Pipe on the rest of the buttercream over the crumb coat, and smooth it with an offset spatula and bench scraper. Pipe dollops of leftover buttercream around the edge of the cake, and sprinkle it with the crushed peppermints.
Store the cake in an airtight container in the fridge for up to 4 days.