Step 1: In a large bowl, cream together the softened butter and granulated sugar until the mixture is light, fluffy, and pale in color (about 3-5 minutes).
Step 2: Add the egg and vanilla extract to the butter mixture and beat until well combined, scraping down the sides of the bowl as needed.
Step 3: In a separate medium bowl, whisk together the all-purpose flour, fine cornmeal, baking powder, and salt.
Step 4: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until the dough just comes together. Do not overmix.
Step 5: Wrap the dough in plastic wrap and refrigerate for at least 30 minutes to allow the cornmeal to hydrate.
Step 6: Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper. Scoop rounded tablespoons of dough and place them 2 inches apart on the sheets.
Step 7: Bake for 10-12 minutes, or until the edges are just lightly golden. Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack.