Prepare the baking sheets. Preheat oven to 350°F / 175°C and place racks on the upper third and lower third of the oven. Line two baking sheets with parchment paper.
Mix dry ingredients. In a large bowl, sift together 1½ cups white pastry (or all-purpose) flour, 1 cup whole wheat pastry (or whole wheat) flour, ½ tsp baking powder, ½ tsp baking soda, and ¾ tsp Kosher salt. Set aside.
Cream the sugar. In the bowl of a stand mixer fitted with paddle attachment, cream together 12 Tbsp room temperature butter and 1¼ cups sugar until light and fluffy, about 2 minutes on medium speed. Add 2 tsp vanilla extract and 2 eggs, one at a time, making sure they are completely mixed in.
Slowly add dry ingredients and buttermilk. Scrape down the sides of the bowl, then with the mixer on low, alternate between adding dry ingredients and the ⅔ cup buttermilk, until both are gone, ending with the flour mixture. The dough will be sticky.
Scoop black and white cookies. Use a ¼ cup cookie scoop to transfer dough to prepared baking sheets, leaving at least 2-inch between cookies. You should be able to fit about 6 cookies per standard ¼ sheet tray.
Bake the cookies. Bake for 9 minutes, rotate the pans, then bake an additional 8 minutes. Do not over bake. Allow cookies to cool on the pan for about 5 minutes, then transfer to a wire rack to completely cool. Repeat with remaining cookie dough.
Make the icing. In a large bowl, whisk together 3½ cups confectioner's sugar, 3 Tbsp cream, 3 Tbsp milk, and 1 tsp vanilla extract. Transfer half of the icing to another bowl and whisk in ½ cup cocoa powder. If either icing is too thick (it needs to be easily spreadable) add milk, 1 tablespoon at a time, whisking until fully incorporated, until desired consistency is reached.
Frost the cookies. Use a small rubber or offset spatula to spread white icing on one half of each cookie. These need to set for about 15-20 minutes. Carefully spread chocolate icing on the opposite side, being careful not to overlap. Enjoy immediately or store in an airtight container.