Using an electric stand mixer with the paddle attachment, cream the butter and sugar together until fluffy.
Add the egg, lime juice, and lime zest and mix until combined.
In a medium bowl, mix the flour, baking soda, and salt.
Add the flour mixture to the butter mix and mix on low speed until incorporated.
Add the corn meal and mix on low speed until just incorporated.
Scoop the dough by the tablespoon and roll into balls. Place two inches apart on a parchment-lined baking sheet and chill in the refrigerator for 30 minutes. (Chill any unused dough at the same time).
Preheat the oven to 350 degrees.
Remove the cookie dough balls from the refrigerator and using a cup, swirl the bottom of the cup in flour, then flatten each cookie into a disk.
Bake 10-12 minutes until the edges are golden brown.
Whisk together the lime glaze ingredients.
Spoon on to the tops of the cooled cookies.
Allow the glaze to set before storing.