Preheat oven to 350°F. Line two baking sheets with parchment paper and set aside.
To make the cookies, cream together the butter and sugar until pale and fluffy, about 3 minutes. Scrape down the sides and mix in the egg and vanilla extract.
In a medium bowl, sift or whisk together the flour, baking soda, and salt.
Add the dry ingredients alternating between the buttermilk, starting and ending with dry ingredients. Make sure to mix until just combined after each addition. Fold in the crushed chocolate sandwich cookies until evenly distributed.
Use a medium cookie scooper (or a small one but double up the portion) to drop mounds of cookie dough onto the prepared baking sheets, making sure to leave at least 2-inches of space between each. Lightly wet your fingers and gently smooth out the cookie dough mounds.
Bake until lightly golden brown and the centers are springy to the touch, 12 to 13 minutes. Remove from the oven and cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.
To make the icing, in a medium bowl, whisk together the powdered sugar, light corn syrup or honey, vanilla and enough milk as needed to create a smooth barely pourable icing. Transfer half to another bowl and whisk in the cocoa powder.
Turn the cookies upside down so the flat side is facing up. Then use a butter knife or small spatula to spread the chocolate icing on half of the cookie to coat evenly. Then coat second half with the vanilla icing, sprinkling with cookie crumbs while still wet. Once all are frosted, allow them to set before serving. Leftovers can be wrapped or stored in an airtight container for up to 3 days Enjoy!