Preheat oven to 350°F. Line a baking sheet pan with parchment paper.
Add sugar and lime zest to a small bowl. Use a fork or your fingers to rub the zest into the sugar for 1-2 minutes. The goal is to extract as much lime zest and infuse the flavor into the sugar.
With an electric mixer, beat the butter and lime-sugar on high until light and fluffy, 3-4 minutes. Add the eggs one at a time, mixing after each, then the vanilla extract.
In a separate bowl, combine the flour, cornmeal, baking powder, and salt. Add to the wet ingredients and mix until just combined.
Place rounded tablespoons of dough (I used a #40 1½ Tbsp cookie scoop) on the prepared sheet pan about 2 inches apart. Gently press each ball with the palm of your hand to flatten slightly.
Bake until the edges are just turning golden brown and centers are still pale, about 17-20 minutes. Do not over-bake. Let cool on the sheet pan for 5 minutes, then remove to a cooling rack to cool completely.
To make the glaze, whisk together the powdered sugar, lime zest, and lime juice. Add milk 1 teaspoon at a time if needed to reach the consistency you want. It should be thin enough to spread over the cookies.
Glaze the cookies on a wire rack with a sheet pan placed underneath to catch excess glaze. Let the glaze harden for several minutes before serving.
These cookies are best served the day they are made, but will keep in an airtight container at room temperature for a few days.