Cream butter and sugar until light and fluffy, about 2 minutes.
Add egg white, vanilla, and peppermint extracts. Mix until thoroughly combined.
Combine flour and cocoa powder, and add to butter mixture ⅓ at a time.
Mix until just combined.
Divide dough in two, and roll out each half to just under ¼ inch thick between two pieces of parchment.
Chill dough for at least 3 hours.
Preheat oven to 325°F.
Cut out cookies using a 1 ½-2 inch cookie cutter, and arrange on a parchment lined pan with at least an inch between each cookie. Reroll and cut dough scraps for another round of baking.
Bake for 12-14 minutes, taking care not to let the cookies burn.
Let cookies cool completely on a cooling rack.
Melt chocolate, shortening, and peppermint extract in a double boiler on the stove at low heat, stirring frequently.
Once completely melted, dip cookies one by one into chocolate mixture, using a fork to drain the excess chocolate before placing on a parchment lined baking sheet.
Continue until all cookies are coated.
Refrigerate cookies until chocolate forms a hard shell, and enjoy.