Preheat oven to 350 degrees. Use parchment paper or a silpat mat to line a baking sheet. Cream the butter, oil, sugar, and brown sugar in a stand mixer until light and fluffy (about 2-3 minutes). Scrape the sides of the bowl to make sure all ingredients are blended.
Add in the egg, vanilla, butter flavoring, and honey and beat until combined.
Add in flour, cornstarch, baking soda, salt, cream of tartar and baking powder. Mix until all combined. Add in corn meal and mix until combined.
Scoop out dough and roll into balls. Scoop out 1 TBSP dough for smaller cookies (or 1/4 cup dough for BIG cookies). Place dough balls on prepared pan (12 cookies to a pan if smaller cookies) and slightly flatten with your hand to about 3/4 – 1 inch thick.
Bake for 8-9 minutes. Cool cookies cookie sheet for 10 minutes before removing to cooling rack. NOTE: If you decide to make the bigger cookies, you’ll need to bake them for 13-15 minutes.
While cookies are cooling, make the frosting.