In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the butter, granulated sugar, and brown sugar together on medium high speed until fluffy and light in color, about 2-3 minutes. Add the egg, vanilla extract, and peppermint extract, and then beat on high speed until combined. Scrape down the sides and bottom of the bowl as needed.

In a separate bowl, whisk the flour, cocoa powder, baking soda, espresso powder, and salt together until combined. With the mixer running on low speed, slowly pour into the wet ingredients. Beat on low until combined and then beat in the chocolate chips. Cover dough tightly and chill in the refrigerator for at least 3 hours and up to 3 days.

Remove cookie dough from the refrigerator and allow to sit at room temperature for 10 minutes. If the cookie dough chilled longer than 3 hours, let it sit at room temperature for about 20 minutes.

Preheat oven to 350°F (177°C). Line large baking sheets with parchment paper or silicone baking mats. Set aside.

Scoop and roll dough, 1.5 Tablespoons of dough each, into balls. Arrange 2-3 inches apart on the baking sheets.

Bake the cookies for 11-12 minutes or until the edges appear set and the centers still look soft. Cool cookies for 5 minutes on the baking sheet, and then transfer to a cooling rack to cool completely.

Melt the chopped white chocolate in a double boiler or use the microwave in 15 second increments, stirring after each increment until completely melted and smooth. Dip each completely cooled cookie halfway into the white chocolate and sprinkle crushed candy canes on top. Place the baking sheet into the refrigerator to help the chocolate set, about 1 hour.