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 3 minutes. (Here’s a helpful tutorial if you need guidance on how to cream butter and sugar.) 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 and salt together until combined. With the mixer running on low speed, slowly pour into the wet ingredients. Beat on low until combined. The cookie dough will be quite thick. Switch to high speed and beat in the milk, then the chocolate chips. The cookie dough will be sticky and tacky. Cover dough tightly and chill in the refrigerator for at least 3 hours and up to 3 days. Chilling is mandatory for this sticky cookie dough.

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. This makes the chilled cookie dough easier to scoop and roll.

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

Scoop and roll dough, about 1 Tablespoon of dough each, into balls. Arrange 2-3 inches apart on the baking sheets. The cookie dough is certainly sticky, so wipe your hands clean after every few balls of dough you shape.

Bake the cookies for 10-11 minutes or until the edges appear set and the centers still look soft. Tip: If they aren’t really spreading by minute 9, remove them from the oven and lightly bang the baking sheet on the counter 2-3x. This helps initiate that spread. Return to the oven to continue baking.

Cool cookies for 5 minutes on the baking sheet, and then transfer to a cooling rack to cool completely. The cookies will slightly deflate as they cool.

With a handheld or stand mixer fitted with a paddle attachment, beat the butter on medium speed until creamy, about 2 minutes. Add confectioners’ sugar, heavy cream/milk, vanilla extract, and peppermint extract. Beat on low speed for 30 seconds, then increase to medium-high speed and beat for 2 full minutes. Taste. Beat in a pinch of salt if frosting is too sweet. I usually add a little less than 1/8 teaspoon. And, if desired, beat in another drop of peppermint extract.

Top each cookie with frosting. Garnish frosting with crushed candy canes. Frosting doesn’t dry completely, but will set after several hours. Carefully stacking frosted cookies is fine after that.

Cover and store cookies at room temperature for up to 1 day or in the refrigerator for up to 1 week.