Adjust oven rack to lower third of oven and preheat to 350°F. Line baking sheets with parchment paper.

Using a stand or hand mixer, cream together butter, sugars, and vanilla.
In a bowl, combine flour, cornmeal, and salt. Add dry ingredients to butter mixture and stir to create a soft dough.

Shape dough into 1-inch balls and space about 1-1/2 inches apart on prepared baking sheets. Bake for 12 to 15 minutes, or until bottoms are lightly colored.

While cookies are still warm on baking sheets, use a small sieve to sprinkle confectioners’ sugar over cookies. When thoroughly cool, dredge them in confectioners’ sugar for a second coating.