Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper and set aside.
In a medium bowl, whisk together the flour, cocoa powder, baking powder, baking soda, and salt. Set aside.
In the bowl of a stand mixer, beat the sugar and butter together on medium-high speed until light and fluffy, about 3 minutes.
Mix in the egg and egg yolk and mix well, scraping down the sides of the bowl.
Add the sour cream, vanilla, and peppermint extract and beat just until combined.
Add the dry ingredients, mixing on low just until combined, scraping down the bowl a few times to make sure the dough mixes well.
Roll 1 ½ tablespoons of dough into a small ball.
Place on the prepared baking sheets, about 2 inches apart, and press it down until it is ¼ to ½-inch thick.
Continue with the rest of the dough.
Bake for 7-12 minutes, then immediately transfer the cookies to a wire rack to cool completely.
To make the frosting, beat the butter, peppermint extract, and vanilla together.
Sift in the powdered sugar and cocoa powder and beat until well incorporated and smooth.
Add in heavy cream and mix until well-combined.
Once the cookies have cooled completely, frost each cookie and top with crushed peppermint candies.
Notes
All-purpose flour: It's important to sift the flour before measuring it. To measure it, lightly spoon into the measuring cup and level.
Cocoa powder: The process for measuring the cocoa powder is the same as for the flour. I haven't tried using Dutch-processed cocoa powder in this recipe.
Butter: I use salted butter. If you're using unsalted butter, increase the salt in the cookies to 1/2 teaspoon and add a couple of pinches of salt to the frosting.
Sour cream: I recommend using full-fat sour cream in this recipe.