Preheat oven to 350 degrees. Line an 8x8 baking pan with foil and lightly coat with baking spray, set aside.
In the bowl of a stand mixer, beat together the melted butter, sugars and vanilla extract for 30 seconds on medium. With mixing speed on low, add the eggs one at a time, mixing until thoroughly combined.
In a medium size mixing bowl, whisk together the flour, sifted cocoa powder, baking powder and salt. With mixing speed on low, gradually add the dry ingredients, mixing until no flour pockets remain. Remove the bowl from the stand and using a spatula, fold in the chopped chocolate, if desired.
Pour batter into the prepared pan, spreading into an even layer. Place in the oven and bake for 18-22 minutes or until set, taking care not to over-bake. Remove from the oven and allow to cool for 30 minutes or until brownies are no longer warm (you can speed up the process by letting them cool uncovered in the refrigerator).
Meanwhile, melt the white chocolate in a double boiler or in the microwave. Allow the chocolate to cool at room temperature for 15 minutes.
In the bowl of a stand mixer fitted or a large mixing bowl using a handheld mixer, beat the butter on high until creamy, about 2 minutes. With mixing speed on low, gradually add the powdered sugar, milk or cream, vanilla extract, peppermint extract, and salt. Mix until combined. Add the white chocolate and 2 tablespoons of crushed candy canes, beat on high for 1 minute.
Using an offset spatula, frost the cooled brownies. Sprinkle with remaining candy cane bits. Store in the refrigerator until ready to serve.