To prepare the cookies: Preheat oven to 350°F. Line sheet pan with parchment paper.
In bowl of stand mixer fitted with paddle attachment, beat together butter and sugar until light and creamy, about 2-3 minutes. Add eggs and yolk one at a time, mixing after each. Add vanilla.
In separate bowl, combine flour, baking powder, baking soda, salt and lemon zest. Add dry ingredients to butter mixture in three additions, alternating with buttermilk (ending with flour mixture), scraping down bowl as necessary.
Use 1/4 cup scoop to drop cookie dough onto prepared sheet pan, leaving 3 inches between each cookie. Bake until edges are golden brown and tops bounce back when touched. Cool on wire rack.
To prepare peppermint icing: Whisk together powdered sugar, heavy cream, milk and peppermint extract in large bowl. If mixture is too thick, add more milk. Pour half of mixture into small bowl to use as peppermint icing.
To prepare chocolate icing: Melt chocolate in heat proof bowl in microwave in 30 second intervals. Stir until smooth. Whisk chocolate into remaining half of peppermint icing. Add milk as needed to achieve same consistency as plain peppermint icing.
To decorate cookies: Turn cookies upside down so flat bottoms face up. Use small offset spatula or knife to spread small amount of vanilla frosting on half of each cookie. Let cookies set 10 minutes, then repeat with chocolate frosting on other half of each cookie. Sprinkle with crushed candy canes. Let set 1 hour before serving.