Using an electric mixer, cream the butter and sugar together until light and fluffy, 3-5 minutes. Then beat in the salt, lime zest, egg, and vanilla. Once blended, slowly add the flour, followed by the cornmeal. Scrape the bowl and beat again for just a few seconds to combine.
Preheat the oven to 375 degrees F (or 350 convection) and line a cookie sheet with parchment paper. Dump the dough out onto large piece of foil and use the foil to shape the dough into a 9 1/2-inch log, about 2 inches thick. Wrap well and freeze for at least 30 minutes to firm up the dough.
When ready to bake, open the foil and sprinkle cornmeal over and around the dough log. Roll it back-and-forth to coat the log on all sides. Then cut the log into 16-18 equal slices. Lay the cookies on the prepared baking sheet 2 inches apart. Bake for 8-10 minutes until just barley golden on the edges, but soft in the middle. (7-8 minutes convection.) Allow the cookies to cool on the cookie sheet completely before moving.