Preheat oven to 350 degrees Fahrenheit. Cover two cookie sheets with parchment paper and set aside.
In a large bowl, combine the oats and chocolate chips, set aside.
In the bowl of your stand mixer, mix the butter and both sugars using the paddle attachment until creamy.
In a medium bowl, sift together the flour, baking soda, salt and cocoa powder. Add this mixture to the butter mixture and beat until thoroughly combined. The batter will be very stiff at this point.
Stir the milk and vanilla into the beaten eggs, and then add this to the stiff butter mixture, beating until thoroughly combined. Add the oat and chocolate chip mixture and stir until well-mixed.
Using a 2-tablespoon scoop, drop batter two inches apart on prepared cookie sheets. Bake for 9-12 minutes, until cooked through. Cool on pan for one minute, then transfer to wire racks to cool completely.
Melt the white chocolate with the peppermint extract and coconut oil in the top of a double boiler (or use a heatproof bowl over a small saucepan of simmering water). Holding a cooled cookie between your thumb and forefinger, dip the edge into the white chocolate to cover the top third of the cookie. Place on a rack over parchment paper and immediately sprinkle on some crushed candy cane pieces (before the white chocolate dries completely.)
Let white chocolate dry completely (this will take an hour or so) and then store in a re-sealable container with pieces of parchment paper in between to keep them from sticking.