Preheat oven to 350°F. Line two baking sheets with parchment paper.
Whisk flour, cornmeal, baking powder, baking soda, and salt in a bowl; set aside.
Cream softened butter with granulated and brown sugars until very fluffy, 2–3 minutes.
Beat in honey, egg, vanilla, and sour cream until smooth and combined.
Add dry ingredients to the bowl and mix on low just until no dry streaks remain.
Scoop 1/4-cup portions of dough, roll into balls, and gently flatten into thick discs.
Bake 10–12 minutes until edges set and tops look matte; centers should be soft.
Cool cookies on the sheet 10 minutes, then transfer to a rack to cool completely.
Beat frosting butter until creamy. Gradually mix in powdered sugar, honey, salt, and vanilla.
Stream in heavy cream until spreadable. Swirl frosting onto cooled cookies and drizzle with honey.