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Combine and sift into a mixing bowl the flours, potato starch, baking soda, xanthan gum, and salt. Add the almond flour, cornmeal, and sugar and mix to combine. Cut in the butter and mix until crumbs form, but do not overmix; you do not want a paste consistency.
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Stir in the dried fruit and lime zest in three batches. Add the egg, vanilla extract, and almond extract and mix well to incorporate.
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Divide dough into three equal portions. Form each portion into a log about 8 inches long and 1 inch wide. Wrap them individually in plastic wrap and freeze for at least four hours or overnight.
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Preheat the oven to 350°F. Line a sheet pan with parchment paper. Cut each frozen dough log into rounds 1⁄4 to 1⁄2 inch thick. Place the rounds 1 inch apart on the sheet pan. Bake until golden brown, 10 to 12 minutes. Let the cookies rest in the pan for 3 minutes, then transfer to wire racks to finish cooling.