Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper and set aside.
In the bowl of a stand mixer fitted with the paddle attachment (or using an electric hand mixer), cream together the softened butter and granulated sugar on medium speed until light and creamy, about 2-3 minutes.
½ cup unsalted butter, 1 cup granulated sugar
Scrape down the bottom and sides of the bowl. Add the slightly chilled egg, mix on medium speed for 20 seconds, then add the cold egg yolk. Scrape down the sides, add vanilla extract, and mix on low until combined.
1 large egg, 1 egg yolk, 1 teaspoon pure vanilla extract
Add the flour, Dutch-processed cocoa powder, baking soda, and salt. Mix on low speed until the dry ingredients are just incorporated. Be careful not to overmix.
1 & ½ cups all-purpose flour, ½ cup Dutch-processed cocoa powder, ¾ teaspoon baking soda, ½ teaspoon salt
Using a 3-tablespoon cookie scoop, form 12 cookie dough balls. Take two dough balls and roll them together to create one large cookie ball. You should end up with 6 large cookies.
Place the cookies on the prepared baking sheet, spacing them evenly. Gently flatten each to about ½ inch in thickness.
Bake in the preheated oven for 10-12 minutes. Allow the cookies to cool on the baking sheet for 10 minutes before transferring them to a cooling rack.
For the Peppermint Frosting:
While the cookies are cooling, make the frosting. In a stand mixer with the paddle attachment, beat the room-temperature butter on medium-high speed until light and fluffy, about 1 minute.
½ cup unsalted butter
Add the powdered sugar, heavy whipping cream, peppermint extract, vanilla extract, and salt. Start mixing on low speed until incorporated, then increase to medium-high and beat for an additional 2 minutes. Scrape down the bowl to ensure even mixing.
2 cups powdered sugar, ½ teaspoon vanilla extract, Pinch of salt, 2 tablespoons heavy whipping cream, ½ teaspoon peppermint extract
Gently fold in the crushed candy canes or peppermint candies using a rubber spatula.
¼ cup crushed candy canes or peppermint candies
Once the cookies have fully cooled, use an offset spatula to spread the frosting evenly over the tops, creating a smooth surface. Place the frosted cookies in the fridge for 15 minutes to set.
For the Peppermint Bark Topping:
While the cookies chill, melt the dark chocolate and butter together in a microwave-safe bowl in 30-second increments, stirring every 30 seconds until smooth.
Remove the cookies from the fridge. Using a large spoon, gently pour the melted chocolate over the frosting on each cookie, spreading it to the edges and smoothing it out with the back of the spoon.
Top with additional crushed peppermint candies, then return the cookies to the fridge for about 1 hour to allow the chocolate to harden.