Beat the softened butter for two minutes, until light and fluffy. Add the sugar, milk, and peppermint extract. Stir until combined, then beat until smooth. Stir in red food coloring until it is the shade you want. Spread the frosting over the cooled brownies. Place in the fridge so the buttercream can chill and harden.

Heat the chocolate chips, butter, and light corn syrup in the microwave at 50% power for 2 minutes and then stir. Continue heating at half power, stirring ever 15 seconds until smooth. Stir in the peppermint extract. Pour the chocolate peppermint icing over the hardened frosting and quickly spread it out. Immediately sprinkle with chopped candy canes, before the icing hardens. Then, return the bars to the fridge so the chocolate can set.

Once chilled, cut the brownies into squares. It helps to heat a knife under hot water, wipe dry and then use the hot knife to easily slice through the chilled layers. Store in the fridge.