Note: This cookie dough requires at least 2 hours of chilling, but ideally overnight. The cookies will be thicker the longer they chill (up to 3 days).
In a medium bowl, whisk the flour, cocoa powder, baking soda, and salt together until combined.
In a large bowl using a hand-held or stand mixer fitted with a paddle attachment, beat the butter, granulated sugar, and brown sugar on medium-high speed for 2-3 minutes, until light and fluffy.
Add the egg, vanilla extract, peppermint extract, and milk and beat until combined, scraping down the sides and bottom of the bowl as needed.
Turn the mixer down to low speed and slowly add the dry ingredients into the wet ingredients until combined.
Add the chocolate chunks and mix using a rubber spatula to incorporate. Cover with plastic wrap and chill for at least 3 hours or up to 3 days. Do not skip this step (it prevents them from spreading).
Preheat the oven to 350°F and line two large baking sheets with parchment paper.
If the dough was chilling for longer than 24 hours, allow it to soften at room temperature for 15-30 minutes for easier scooping. Using a large cookie scoop (3-4 Tablespoons) scoop the cookie dough and drop them 2-3 inches apart on the prepared baking sheet.
Bake the cookies in the preheated oven for 10-12 minutes or until the edges are set and the tops look slightly soft. Cool the cookies for 5 minutes on the cookie sheet then transfer them to a cooling rack to cool completely.
Before dipping in white chocolate, make sure the chocolate chips inside the cookies are fully cooled and set (Note 2). Place the white chocolate and coconut oil in a tall and skinny microwave safe bowl, jar, or measuring cup. Microwave in 30 second increments, stirring between each, until it's melted and smooth.
Dip half of each cookie in the melted white chocolate then place it on a piece of parchment paper and sprinkle with crushed candy canes. Cool completely until the white chocolate is set. Baked cookies stay fresh in an airtight container at room temperature for 5 days or can be frozen for up to 3 months.