Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone baking mat.

Cream the fats and sugars: In a large mixing bowl, beat together the softened butter, vegetable oil, granulated sugar, and brown sugar using a hand mixer or stand mixer. Blend until the mixture is light and fluffy, approximately 2 to 3 minutes.

Add wet ingredients: Incorporate the egg, vanilla extract, butter extract, and honey. Mix until fully combined.

Combine dry ingredients: In a separate bowl, whisk together the flour, cornstarch, baking soda, baking powder, cream of tartar, cornmeal, and salt.

Form the cookie dough: Gradually add the dry ingredients to the wet mixture, mixing on low speed until just combined. Avoid overmixing to ensure a soft texture.

Portion the dough: Using a large cookie scoop (approximately 3 tablespoons per scoop), portion out the dough and place it on the prepared baking sheet, spacing each cookie about 2 inches apart.

Bake: Bake the cookies in the preheated oven for 8 to 10 minutes, or until the edges are lightly golden and the centers are set. Do not overbake.

Cool: Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.

Prepare the frosting: In a medium mixing bowl, beat the softened butter until smooth. Gradually add the powdered sugar, mixing well. Stir in the honey and vanilla extract. Add heavy cream, a small amount at a time, until the desired frosting consistency is achieved.

Frost the cookies: Once cooled, pipe or spread a generous amount of honey buttercream onto each cookie. Optionally, drizzle a small amount of honey on top for garnish.