For the cookies:
1 large egg yolk
1/2 teaspoon peppermint oil
1/2 teaspoon vanilla extract
1 3/4 cups all-purpose flour
1 cup powdered sugar
1/2 cup unsweetened Dutch-process cocoa powder
1/4 teaspoon fine salt
1 cup unsalted butter (2 sticks), chilled and cut into small pieces
For the coating:
Ice
2 pounds semisweet chocolate couverture, separated into 1 (24-ounce) portion and 1 (8-ounce) portion
To make the cookies:
1 Place the egg yolk, peppermint oil, and vanilla extract in a small bowl and whisk to break up the yolk.
2 Combine the flour, sugar, cocoa powder, and salt in the bowl of a food processor fitted with a blade attachment and pulse a few times to aerate and break up any lumps. Add butter and pulse until the mixture looks like sand, about 25 (1-second) pulses. Add yolk mixture and pulse just until the dough forms into a ball, about 15 (1-second) pulses.
3 Turn the dough out onto a clean work surface and roll into 2 logs, each about 1 1/2 inches in diameter. Wrap the logs in plastic wrap and refrigerate until just firm but still pliable, about 1 hour. (The logs will flatten slightly while chilling. If you have a paper towel tube available, cut it in half lengthwise and nestle the cookie dough in there; this will help the dough keep its cylindrical shape while it chills.) Reshape the logs so they are perfectly round and refrigerate until firm, about 1 hour more.
4 Heat the oven to 350°F and arrange a rack in the middle. Line a baking sheet with parchment paper; set aside.
5 Remove a dough log from the refrigerator, remove the plastic wrap, and slice the dough into 1/8-inch coins. Place the cookies 1/2 inch apart on the prepared baking sheet. (About 30 cookies will fit on 1 sheet.) Rewrap the extra cookie dough in plastic and refrigerate until ready to bake the second batch.
6 Bake the cookies until the edges are firm but the tops are still soft, about 9 to 11 minutes. Remove to a wire rack to cool completely. Repeat with the remaining dough. (You can use the same piece of parchment paper.)
To make the coating and finish:
1 Line two baking sheets with parchment paper; set aside.
2 Prepare an ice water bath by filling a large bowl with 2 inches of cold water and adding 3 to 4 ice cubes; set aside.
3 Bring a medium saucepan filled with 1 to 2 inches of water to a simmer over high heat; once simmering, turn off the heat. Place 24 ounces of the chocolate couverture in a dry, heatproof bowl. Set the bowl over the saucepan and stir until the chocolate is completely melted and has reached 118°F. (Make sure the chocolate does not come in contact with water or exceed 120°F. If either happens, start over, as the chocolate will no longer be usable.)
4 Remove the bowl from the saucepan. Add the remaining 8 ounces chocolate and stir until all the chocolate has melted and the temperature has cooled to 80°F. To speed up the cooling process—but only after all of the chocolate has melted—place the bowl over the reserved ice water bath.
5 Once cooled, return the bowl of chocolate to the saucepan and stir until the chocolate reaches 88°F; immediately remove from heat. Do not remove the thermometer from the bowl; check the temperature periodically to make sure it stays between 87°F and 89°F. (The chocolate must remain in this temperature range while dipping the cookies or it will not set properly.) Keep the saucepan of water over low heat and, when needed, set the bowl of chocolate over it to reheat.
6 To test if the chocolate is properly tempered, spread a thin layer on parchment paper and place it in the refrigerator for 3 minutes to set. If the chocolate hardens smooth and without streaks, it is properly tempered. (If it is not properly tempered, let the melted chocolate harden and start the tempering process over again: Bring the chocolate up to 118°F, then down to 80°F, then up again to 88°F.)
7 Spoon the tempered chocolate over a cookie and set on a wire rack to set. Repeat until all the cookies have been coated. Let sit at room temperature until completely set, about 20 minutes. Store in an airtight container at room temperature for 1 week or in the freezer for up to 1 month.