In a mixing bowl, beat butter for about 30 seconds with an electric mixer at medium speed. Add sugar and beat on medium speed until fluffy, about 1 minute. Add egg, vanilla extract, and peppermint extract and beat until well combined, scraping down the sides of the bowl as needed.
In a separate bowl, whisk together flour, cocoa powder, baking powder, and salt. With the mixer on low speed, gradually add flour mixture to the butter mixture, beating until just combined.
Divide dough into two equal parts and use your hands to form each part into a disc. Wrap in plastic wrap and refrigerate for 1 hour. It may be tempting to skip the chilling time, but chilling the dough is essential to keeping the cookies from spreading while baking.
Working with one disc at a time, remove from the refrigerator and roll to 1/4-inch thickness. When rolling the dough, sprinkle cocoa powder on your work surface and rolling pin instead of flour to avoid creating white spots on your cookies.
Use a 2-inch round cookie cutter to cut the dough into circles. Transfer cut circles to a baking sheet with a small spatula. Re-roll the remaining dough scraps and continue to cut circles until all the dough is used.
Bake cookies at 350 degrees for 8 minutes. The cookies will still appear soft. Allow to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Place chopped chocolate in the top of a double boiler over simmering water. Add canola oil. Heat, stirring frequently, until chocolate is completely melted and smooth. Remove from heat and stir in peppermint extract.
One at a time, dip each cooled cookie in chocolate. Place each cookie in the melted chocolate, push down with a fork to submerge completely, and use the fork to lift the cookie out. Gently tap the fork on the side of the bowl to allow excess chocolate to drip off. Place dipped cookies on a baking sheet lined with parchment or waxed paper.
Allow cookies to stand until chocolate is completely set.