In a large bowl using a handheld mixer or stand mixer fitted with a paddle attachment, beat the butter and granulated sugar together on medium speed until completely smooth and creamy, about 2 minutes. Beat in the egg and vanilla extract on high speed. Scrape down the sides and up the bottom of the bowl and beat again as needed to combine.

Whisk the flour, cocoa powder, baking powder, and salt together in a medium bowl. On low speed, slowly mix into the wet ingredients until combined.

Divide the dough into 2 equal parts. Dust your rolling pin, cookie dough, and work surface with cocoa powder. Roll each portion out to about 1/4″ thickness on a piece of parchment paper or silicone baking mat. Stack the pieces (with parchment paper between) onto a baking sheet and refrigerate for at least 1 hour. Chilling is mandatory so the cookies hold their shape. If chilling for more than a couple hours, cover the top dough piece with a single piece of parchment paper. You can chill up to 2 days.

Once chilled, preheat oven to 350°F (177°C). Line 2-3 large baking sheets with parchment paper or silicone baking mats. Remove one of the dough pieces from the refrigerator and using a 2.5-inch circle cookie cutter, cut in shapes. Transfer the cut cookie dough to the prepared baking sheets. Re-roll the remaining dough and continue cutting until all is used.

Bake for 10 minutes. The cookies will still appear soft in the centers. Cool cookies on a baking sheet for 5 minutes, then transfer to a wire rack to cool completely before coating.

Line 2 baking sheets with parchment paper or silicone baking mats. Melt the chopped white chocolate and oil together in a double boiler or in the microwave in 15-second increments, stopping and stirring after each until completely smooth. Add another drop of oil to thin out if needed. After melting, stir in 1/2 teaspoon of peppermint extract. Taste. Add 1/4 teaspoon more if desired. (I suggest using 3/4 teaspoon because when the white chocolate sets, the peppermint flavor won’t be as strong.) Drop 1 cookie into white chocolate. Using a 2- or 3-pronged candy dipping tool, flip the cookie over to coat all sides. Lift the cookie up out of the chocolate and gently tap the dipping tool against the side of the bowl so excess white chocolate drips off. Slide coated cookie off the dipping tool onto prepared lined baking sheet. Sprinkle with crushed candy canes. Repeat with remaining cookies.

Allow chocolate to set completely in the refrigerator for 45 minutes or at room temperature for 90 minutes.

Coated cookies stay fresh covered at room temperature for 3 days or in the refrigerator for up to 10 days.