In the bowl of a stand mixer combine the butter and sugar. Beat with a paddle attachment until light and fluffy. Add the eggs and peppermint oil and beat until fully combined. Scrape down the sides of the bowl and continue beating until the mixture is completely blended.
Add the sour cream, beating to combine. Then add the buttermilk, mixing until the wet ingredients are fully combined. The mixture may appear curdled from all the extra liquid but this is normal.
Add 5½ cups of the flour along with the cornstarch, baking powder, baking ammonia, and salt and beat just until combined. The dough will be soft and sticky but it should hold it's shape. Add another ½ cup flour if the dough is too soft.
Transfer the dough to a mixing bowl and cover with film wrap. Chill at least 1 hour or for up to a day.
Preheat the oven to 350°F / 204°C. Turn out half of the dough onto a floured counter. Sprinkle with a little more flour and gently pat out into a smooth ball. Roll the dough out to ½ inch thickness.
Cut the dough out with a 1 ½ or 2 inch cutter and transfer to a parchment lined baking sheet. If possible, chill the baking sheet for 5 to 10 minutes before baking.
Bake the cookies for 7 to 9 minutes or until the cookies are puffed and spring back when lightly touched on top. Take the cookies out before the edges turn brown. Transfer the cookies to a rack to cool. If desired, top with icing.
Make the Icing
In a small bowl whisk together the icing sugar, peppermint oil, and enough milk to make a thin, spreadable icing.
Spread a thin layer of icing on top of the cooled cookies, sprinkle with shredded coconut if desired and enjoy!
Notes
*Substitute with 2 tsp peppermint extract if desired