For the cookies: Line a baking sheet with a silicone liner or parchment paper. Set aside.

In a food processor, pulse together the butter, sugar, and lime zest until combined. Add the egg, lime juice, and vanilla. Process until smooth. Add the rice flour, cornmeal, baking powder, and salt. Process until the mixture forms a dough. Place the dough on a lightly floured surface and knead for 20 seconds. Form the dough into a ball, wrap in plastic, and refrigerate for 20 minutes.

Place an oven rack in the center of the oven. Preheat the oven to 350 degrees F. Lightly flour a work surface. Cut the dough in half and roll out each piece into a 9-inch diameter circle, about 1/4-inch thick. Using a 3-inch round cookie cutter, cut the dough into 10 circles and arrange on the prepared baking sheet. Gather any scraps of dough, knead together, and roll out to 1/4-inch thick. Cut out 2 additional rounds of dough and place on the baking sheet. Bake until light golden around the edges, 15 to 20 minutes. Cool for 10 minutes and transfer to a cooling rack to cool completely, about 15 minutes.

For the glaze: In a medium bowl, whisk together the powdered sugar, lime zest, and lemon juice until smooth. Spread about 1 teaspoon of the glaze onto each cookie leaving, a 1/4-inch border around the edge. Allow the glaze to harden for about 45 minutes before serving.