Preheat the oven: Set the oven to 350°F (175°C) and line baking sheets with parchment paper.
Cream the sugars and fats: In a large bowl, beat together granulated sugar, brown sugar, butter, and shortening until light and fluffy.
Add eggs and flavorings: Beat in the eggs one at a time, then mix in honey and vanilla extract.
Combine dry ingredients: In a separate bowl, whisk together baking powder, salt, cornmeal, and flour.
Form the dough: Gradually add the dry ingredients to the wet mixture, stirring until a soft dough forms.
Shape the cookies: Scoop or roll dough into balls and place on prepared baking sheets. Flatten slightly with your fingers or the bottom of a glass.
Bake: Bake for 10–12 minutes, or until edges are lightly golden. Let cool on the baking sheet for a few minutes, then transfer to a wire rack to cool completely.
Prepare the frosting: Beat together powdered sugar, butter, honey, and heavy cream until smooth and fluffy.
Frost the cookies: Spread frosting over cooled cookies and serve. Store in an airtight container.