Preheat the oven to 325°F. Line a baking sheet with parchment paper. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, mix the butter, shortening, granulated sugar, brown sugar, eggs, honey, and vanilla.
To the butter mixture, add the flour, cornmeal, cornstarch, salt, baking powder, and baking soda. Mix until the ingredients are incorporated.
Using a ¼ cup (or 2 scoops of dough with a 2 tablespoon scoop), drop portions of dough onto the lined cookie sheet, about 2 inches apart. (If using two 2-tablespoon scoops, combine the scoops of dough into one large cookie.)
Using your hand, gently flatten each ball of dough.
Bake for 14-16 minutes. (Do not overbake these cookies. They are perfect when slightly under-done. And over-baked cookie will be dry. Test the first batch to make sure 14 minutes is right in your oven.)
Once baked, remove the cookies from the oven and allow them to cool on a wire rack while you prepare the frosting.