In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the butter on medium speed until creamy, about 1 minute. Add the confectioners’ sugar and beat on low speed for 30 seconds, then increase to medium-high speed and beat for 2 full minutes, until light and fluffy. Scrape down the sides and bottom of the bowl as needed. (Here’s a helpful tutorial if you need guidance on how to cream butter and sugar.) Add the egg yolk, vanilla extract, and peppermint extract and beat on high speed until combined. Scrape down the sides and bottom of the bowl as needed.

Add the flour, cornstarch, baking powder, and salt. Beat on low speed until combined. The cookie dough will be thick and creamy. Cover dough tightly and chill in the refrigerator for at least 30 minutes and up to 3 days. Chilling is mandatory for this creamy cookie dough.

Preheat oven to 350°F (177°C). Line large baking sheets with parchment paper or silicone baking mats. Set aside.

Scoop and roll dough into balls, about 1 Tablespoon or 20–22g each. Place dough balls 3 inches apart on the baking sheets, and then flatten each slightly with the palm of your hand; they should be just under 2 inches in diameter.

Bake the cookies for 13–14 minutes or until the edges appear set. The cookies won’t darken/brown, so be careful not to over-bake them.

Remove from the oven and allow cookies to cool on the baking sheet for 10 minutes, then transfer to a cooling rack to cool completely before frosting.

With a handheld or stand mixer fitted with a paddle attachment, beat the butter on medium speed until creamy, about 1 minute. Add the confectioners’ sugar, milk, vanilla extract, and peppermint extract. Beat on low speed for 30 seconds, then increase to medium-high speed and beat for 2 full minutes. Taste. To thicken the buttercream, beat in more confectioners’ sugar. To thin out the buttercream, beat in more milk. To help offset more sweetness, beat in a pinch of salt.

If you’d like to tint the frosting, beat in 1 tiny drop of food coloring. Add more to deepen the color, if desired. Using a knife or icing spatula, spread frosting onto the cooled cookies. Sprinkle each lightly with crushed candy canes.

Cover and store frosted cookies at room temperature for up to 1 day, or in the refrigerator for up to 5 days. Frosting “soft-sets” on the cookies after a few hours, meaning it will slightly dry on top.