Preheat oven to 350°F. Line an 8″x8″ square pan with parchment paper, ensuring it overhangs on two sides for easy removal. If concerned about the parchment moving, consider lightly greasing the pan before placing the parchment.
In a medium-sized microwave-safe mixing bowl, melt the butter. Whisk in oil, granulated sugar, eggs, vanilla extract, peppermint extract, and salt until the mixture is light and fluffy (about 1 minute of vigorous whisking).
Switch to a rubber spatula to fold in cocoa powder, chocolate chips and flour until there are no more flour streaks remaining.
Pour the brownie batter into the prepared baking dish. Use the back of a spoon or a small offset spatula to spread the batter evenly to the edges of the pan.
Bake for 35-38 minutes or until a toothpick inserted into the center comes out with moist crumbs. Allow the brownies to cool in the pan for about 10 minutes, then use the parchment paper overhangs to transfer them to a wire rack to cool completely.
For the peppermint frosting:
While the brownies are cooling, make the frosting. In a stand mixer with a paddle attachment, beat room temperature butter until light and fluffy (about 1 minute).
Add powdered sugar, heavy whipping cream, peppermint extract, vanilla extract, and salt. Mix on low until incorporated, then turn on medium-high and beat for 2 minutes. Scrape the bottom of the bowl with a rubber spatula to ensure even mixing. With a rubber spatula, fold in the candy cane pieces.
Once the brownies are cooled, spread the frosting evenly over them, creating a smooth surface. Place back into the fridge for 15 minutes.
For the peppermint bark:
Once the frosted brownies have chilled, melt dark chocolate and butter together in a microwave-safe bowl in 30-second increments, stirring every 30 seconds until smooth.
Pour the warm melted chocolate mixture over the frosting, spreading it to the edges and smoothing evenly with the back of a spoon.
Top with crushed peppermint candies and allow to harden in the fridge for about 1 hour.
Once the peppermint bark has set, use the parchment paper to lift the entire creation from the pan. Place it on a cutting board and cut into squares for serving.