In a large bowl, beat together the butter, brown sugar and granulated sugar until fluffy. There should be no lumps of butter or brown sugar.
Beat the egg, vanilla extract and peppermint extract into the dough.
Sift together the flour, cocoa, baking soda and salt in a separate bowl and whisk together. Then add the dry ingredients to the butter mixture. Beat together until combined. The dough will be thick and sticky.
Stir in the ¾ up semi-sweet chocolate chips.
Cover the bowl with plastic and refrigerate for at least 2 hours or up to 48 hours.
When ready to bake, preheat the oven to 350℉ (180℃) and line cookie sheets with parchment paper or silicone baking mats. If the dough has been chilling overnight, take it out of the fridge and let it sit on the counter for about 10 minutes so that it's easier to work with. If it's only been chilling for 2 hours, you likely don't need to let it sit on the counter to warm up slightly first.
Form the dough into balls with about 1 to 1½ tablespoons of dough each. Place about 2 inches apart on the cookie sheets. You can flatten the dough balls down slightly.
Bake 1 sheet at a time for about 9-11 minutes, or until the tops look just set.
Cool the cookies on the cookie sheet.
If using baking chocolate or chocolate bars for the topping, first finely chop so that the chocolate is in very small pieces. To melt the chocolate, use a double boiler over low gently simmering water (ensure that the bottom of the bowl containing the chocolate does not touch the water underneath). Alternatively, microwave on medium power (not high - which is often the default on your microwave) for 45 second intervals, stirring between each interval. After the first interval, the chocolate will likely still be firm - do not be tempted to increase the power or duration of the intervals - otherwise the chocolate can become grainy.
Drizzle the top of each cookie with chocolate, or paint 1/2 with melted chocolate (I use the back of a spoon). Sprinkle with crushed candy canes.