Preheat oven to 350°F. Line two cookie sheets with parchment paper or silicone baking mats.
Cream butter and both sugars using an electric mixer until fluffy. Mix in egg, baking soda, salt and vanilla. Add cornmeal and flour and mix until smooth.
Place a bit of granulated sugar in a small bowl.
Scoop 2 tablespoon balls of cookie dough and roll them between your palms. Roll the cookie dough in the granulated sugar and place 2-inches apart on cookie sheet. Flatten cookies with the back of a measuring cup or glass (if the dough sticks, spray the glass with nonstick cooking spray).
Bake for 10-14 minutes or until lightly golden around the edges. Cool before removing from cookie sheets.
Store in an airtight container for up to 3 days or freeze for up to a month.