Melt the chocolate in the microwave for 25 second increments, stirring after each until completely melted. Set aside to slightly cool.
In a medium bowl, combine the flour, cocoa powder, espresso powder, baking powder, and salt.
Using a handheld or stand mixer, beat the butter, brown sugar, and granulated sugar together on medium-high speed until smooth and creamy. Add the eggs, vanilla extract, and peppermint extract. Beat on high for 2 minutes. Pour in the melted chocolate and mix on medium-high speed for 2 minutes, until well combined.
Add the dry ingredients and beat on low until combined.
Preheat oven to 350°F. As the oven preheats, cover and chill the dough in the refrigerator for 20 minutes.
Roll or scoop the dough into balls. Make for 12 minutes.
Put the cookies on a cooling rack. Melt the white chocolate, then using a fork, drizzle on the cookies. Add the crushed candy cane topping.