Preheat oven to 350º and line baking sheets with parchment paper.
With an electric mixer, cream the butter and sugar on medium speed for 5 minutes or until light and fluffy. Add the lime zest and beat for 1 minute. Add eggs and vanilla and continue to beat for 2-3 minutes until combined.
In a small bowl, whisk the flour, cornmeal, baking powder and salt. Slowly add the flour mixture to the butter mixture and mix until evenly combined.
Using a small ice cream scoop, scoop onto baking sheet. Flatten each slightly with the palm of your hand.
Bake for 25 minutes or until the cookies are pale brown on the edges and still pale in the center. Let cool for 15-20 minutes. Before cooled completely, brush the cookies with a thin layer of the glaze. and let set for about 10 minutes.
Glaze: Whisk together all ingredients until smooth.