Line two baking sheets with parchment paper and set aside. In a mixer, beat the room temperature butter and granulated sugar together for 1-2 minutes until light and fluffy—this incorporates air and creates a tender cookie texture. Add the egg and vanilla extract, mixing until fully combined. Gradually add the flour while mixing on low speed until the dough just comes together into a ball; be careful not to overmix, as this can make the shortbread tough. Scoop the dough into 2-tablespoon portions and roll into balls, then place them on the prepared baking sheets about 2 inches apart. Gently press each ball into a ½-inch thick disc using the bottom of a glass or your palm. Refrigerate the shaped dough for 15 minutes to relax the gluten and help the cookies maintain their shape during baking.
While the dough chills, preheat your oven to 350°F. Once the dough has finished chilling, bake the cookies for 10-11 minutes until the edges are just set but the centers remain slightly pale—they should not be golden brown, as shortbread bakes quickly and continues to cook slightly as it cools. Remove from the oven and let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely. This resting time on the hot baking sheet helps them firm up without becoming brittle.
While the cookies cool, combine the chopped semi-sweet chocolate and heavy cream in a microwave-safe bowl. Microwave the mixture in 1-minute intervals, stirring well after each interval, until the chocolate is mostly melted and the mixture is smooth and glossy. If needed, microwave for an additional 15 seconds and stir again—I prefer to stop just before the chocolate is fully melted and let the residual heat finish the job, which prevents the chocolate from becoming grainy or burnt. The ganache should be warm enough to dip but not so hot that it slides off the cookies.
Once the cookies are completely cooled and the ganache has thickened slightly (it should coat a spoon but still be pourable), dip the top of each cookie into the ganache, allowing excess to drip off. Immediately sprinkle the finely crushed candy canes over the chocolate while it's still wet—working quickly ensures the toppings stick properly. I like to work in batches so the ganache stays at the right consistency for dipping. Place the dipped cookies on a clean parchment-lined baking sheet and refrigerate for about 20 minutes until the ganache is completely set. Once firm, the cookies can be stored in an airtight container at room temperature.