Beat together 4 ounces unsalted butter, ⅓ cup granulated sugar, ¼ teaspoon peppermint extract, and ½ teaspoon pure vanilla extract in a mixing bowl until combined and fluffy.
Sift 1 cup all-purpose flour and ¼ cup Dutch-processed cocoa powder into a small mixing bowl. Stir in ⅛ teaspoon kosher salt. Add the flour mixture to the butter mixture. Mix until a dough forms, scraping the side of the bowl as necessary while mixing to ensure the flour is mixed well with the butter.
Scrape the dough out of the bowl onto a piece of plastic wrap. Flatten it into a disk, wrap well, and refrigerate for at least 30 minutes and up to 1 hour.
When the dough is firm but still malleable, remove it from the refrigerator and roll it into a ¼ inch thick on a cocoa-dusted surface. Cut with a 2-inch cookie cutter. Reroll the scraps and keep cutting out cookies until the dough is gone.
Place the cookies in a container or a lined baking sheet and freeze for about 15 minutes.
Preheat the oven to 350 degrees Fahrenheit. When the oven is ready, place cookies on another parchment-lined baking sheet 1-2 inches apart and bake for 15 minutes.
Remove from the oven, and let cookies cool on the baking sheet for 5-10 minutes before transferring to a wire rack to cool completely.
Combine 8 ounces chocolate melting wafers, ¼ teaspoon coconut oil, and ¼ teaspoon peppermint extract in a heatproof bowl, set it over a pot of simmering water, and stir until the chocolate is completely melted.
Dip cooled cookies in the melted chocolate using a fork, then turn to coat both sides. Tap excess chocolate off and place coated cookies on a parchment-lined baking sheet. Allow the chocolate coating to dry completely at room temperature or refrigerate to speed it up.