Preheat oven to 350°F. Line a sheet pan with parchment paper.
In the bowl of a stand mixer fitted with a paddle attachment, beat together the butter and sugar until light and creamy, about 2-3 minutes. Add the eggs and yolk one at a time, mixing after each. Add the vanilla.
In a separate bowl, combine the flour, baking soda, baking powder, salt, and lemon zest. Add the dry ingredients to the mixer in 3 additions, alternating with the buttermilk (ending with the flour mixture), scraping down the bowl as necessary.
Use a ¼ cup scoop to drop the cookie dough on the prepared sheet pan, leaving 3 inches between each cookie. You should get 6 cookies on each sheet.
Bake until the edges are golden brown and tops bounce back when touched, about 10 minutes. Cool on a wire rack.
To make the frosting, whisk the powdered sugar, milk, cream, and vanilla together in a bowl. If the mixture is too thick, add more milk. Pour half of the frosting into a small bowl, this will be the vanilla frosting.
To make the chocolate frosting, melt the chocolate in a heat proof bowl in the microwave in 30 second intervals. Stir until smooth. Whisk into the remaining half of frosting. Add water as needed to achieve the same consistency as the vanilla frosting.
To frost the cookies, turn them upside down so that the flat bottoms face up. Use a small offset spatula or knife to spread a small amount of vanilla frosting on half of each cookie. Let the cookies set for 10 minutes, then repeat with the chocolate frosting on the other half of each cookie. Let set for 1 hour before serving.
These cookies are best served the day they are made, but will store in an airtight container at room temperature for a few days.