Preheat oven to 350˚F. Line two baking sheets with parchment or lightly grease.
In the bowl of a stand mixer fitted with a paddle attachment or a large bowl, beat together the butter, brown sugar, and granulated sugar until light and creamy.
Add the eggs, one at a time, followed by the vanilla and peppermint extracts.
In another bowl, combine the flour, cocoa powder, cornstarch, baking soda, and salt. Slowly add to the butter mixture until just incorporated and no streaks remain. Fold in the chocolate chips.
Use a small scoop or spoon to transfer mounds of dough, about 2 tablespoons each, onto the prepared baking sheets, roughly 2 inches apart. Flatten each piece of dough slightly with the palm of your hand and bake in preheated oven until just set, about 10 minutes. Allow to cool on baking sheet for 5 minutes before removing to wire rack to cool completely.
Using a double boiler or a microwave in 20 second increments, melt the 1/2 cup chocolate chips until smooth. Transfer to a pastry bag fitted with a small round tip or a quart sized ziploc bag with a small corner snipped off. Drizzle over the room temperature cookies. Quickly sprinkle the tops with the crushed candy canes before the chocolate has a chance to set.
Once the drizzled chocolate has set, store the cookies in an airtight container for up to 3 days.