The cookie dough must chill for at least 3 hrs. Do not skip this step.
In a large bowl using a stand mixer fitted with the paddle attachment, beat the butter and sugars on medium to medium high speed until fluffy, 1-2 minutes. Beat in egg, vanilla and peppermint extracts.
In a separate bowl, whisk the flour, cocoa powder, baking soda, espresso powder, and salt together until combined. On low speed, slowly mix into the wet ingredients until combined. Add in the chocolate chips. The cookie dough will be a little sticky. Cover dough with plastic wrap and chill for at least 3 hours. Do not skip the chilling, it is an essential step.
Preheat oven to 350° F. Remove cookie dough from the refrigerator. Using a tablespoon scoop out the dough and roll into balls. Place on a cookie sheet that is lined with parchment paper or a silicone mat. If dough is to hard let sit for 15 minutes to make it easier.
Bake the cookies for 8-9 minutes. The baked cookies will look soft in the centers when you remove them from the oven. Do not over bake. Cool for 5 minutes on cookie sheet and then transfer to a cooling rack.
Melt the chopped white chocolate in a double boiler or use the microwave. For the microwave, place the white chocolate in a heat-proof bowl. Melt in 20 second increments, stirring after each increment until melted and smooth. Dip each cookie halfway into the white chocolate and place onto a parchment or silicone mat-lined baking sheet. Sprinkle the crushed candy canes on top of the chocolate. To help set the chocolate place the baking sheet into the refrigerator.