Line a baking sheet with parchment paper. In a microwave-safe bowl, melt the semi-sweet chocolate in the microwave until smooth. Microwave the white chocolate in a separate bowl.
Pour the melted semi-sweet chocolate onto the lined baking sheet, and spread into a thin layer that's about 7 by 10 inches. Drop spoonfuls of the white chocolate evenly over the semi-sweet chocolate. Use a butter knife to swirl the white chocolate through the semi-sweet chocolate. Once swirled, gently tap the baking sheet on the counter until all of the chocolate settles and smooths out.
Sprinkle the crushed candy canes evenly over the chocolate, then place into the refrigerator to set up and harden--about 10 minutes. Once set, peel the chocolate bark off of the parchment and break into pieces. Store in the refrigerator until ready to decorate the cupcakes.
Make the Cupcakes:
Preheat oven to 350ºF degrees, and line a muffin pan with cupcake liners. Add the espresso powder to the hot water, and stir until fully dissolved; set aside.
In a large mixing bowl, whisk together the flour, cocoa powder, baking soda, salt, and both sugars until well combined. Add the oil and yogurt and continue stirring with a whisk. Batter will be thick so you’ll have to add half of the espresso liquid to stir more easily. Add the rest of the espresso liquid, the egg, vanilla, and peppermint extract, stirring until just combined.
Pour batter into cupcake liners, filling 2/3 of the way. Bake in the oven for 16-18 minutes, or until a toothpick inserted in the center comes out with few crumbs attached. Allow cupcakes to cool in the pan for 5 minutes before transferring to a wire rack to cool completely.
Make the Frosting:
Place the chocolate in a microwave-safe bowl and heat in the microwave until smooth and melted. Set aside to cool slightly***.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, salt, vanilla and peppermint extract on high until creamy, about 2 minutes. Add the powdered sugar, a half cup at a time, until fully incorporated. Mix on high for 3 minutes.
Stop the mixer and use a rubber spatula to scrape down the sides and bottom of the mixing bowl. Add the melted chocolate and beat frosting on high for 2-3 minutes. Add the heavy cream, and continue to beat until light and fluffy.
Pipe or spread frosting onto cupcakes, then garnish with crushed peppermints and a piece of peppermint bark. Serve and enjoy!
Notes
*I added 1/4 teaspoon peppermint extract which was just strong enough for me.**If using white chocolate chips which don't melt as smoothly, you might want to add 1-2 teaspoons of vegetable/coconut oil to your chocolate when heating in the microwave. This will result in a much smoother consistency, thus making it easier to incorporate into the frosting.***Make sure the white chocolate doesn't start to harden up again before you add it to the frosting. If it's already begun to set along the edges of the bowl, it'll harden once it hits the frosting, causing chunks of white chocolate throughout. If necessary, reheat chocolate in the microwave at 10-second increments just until it's completely liquid-smooth, but not too warm.