Cream sugar and butter: Preheat oven to 350°F with racks in top third and lower third positions. Line 2 large rimmed baking sheets with parchment paper; set aside. Beat brown sugar, butter, granulated sugar, honey, and oil with an electric mixer on medium speed until until light and fluffy, about 3 minutes.

Add eggs: Add eggs, one at a time, beating well after each addition. Scrape down sides and bottom of bowl, and beat in vanilla until just combined.

Add dry ingredients: Whisk together flour, cornmeal, cornstarch, baking powder, and salt in a medium bowl until no lumps remain. With mixer on low speed, gradually add flour mixture to sugar mixture in 3 additions, beating just until combined after each addition, and scraping down sides of bowl as needed.

Scoop cookies: Using a measuring cup or large scoop, scoop about ⅓ cup of batter onto prepared baking sheets, about 5 cookies per baking sheet, leaving about 4 inches in between each cookie.

Bake cookies: Bake in preheated oven until edges are golden brown and centers are still soft, about 13 minutes, rotating baking sheets between top and bottom racks halfway through bake time. Let cookies cool on baking sheets for 5 minutes; transfer cookies to a wire rack to cool completely.

Prepare the Honey Butter Frosting: Beat butter, honey, and vanilla with an electric mixer on medium speed until light and fluffy, about 2 minutes. Gradually beat in powdered sugar until fully combined and no lumps remain, about 4 minutes, stopping to scrape down sides of bowl as needed.

Add Frosting to Cookies: Top each Cornbread Cookie with about 1 ½ tablespoons of Honey Butter Frosting; using an offset spatula, spread Frosting into an even layer. Drizzle the tops of each frosted Cookie with additional honey, and sprinkle with flaky sea salt. Store Cookies in an airtight container up to 3 days.