To prepare cookies, weigh or lightly spoon white rice flour, potato starch, and tapioca flour into dry measuring cups; level with a knife. Sift together white rice flour, potato starch, tapioca flour, baking powder, and xanthan gum.
Place granulated sugar and butter in a medium bowl; beat with a mixer at medium speed until light and fluffy. Add 1 1/2 teaspoons vanilla, 1 teaspoon milk, and egg, beating until blended. Add flour mixture, stirring until blended. Shape dough into a ball; wrap in plastic wrap. Chill 1 hour.
Preheat oven to 375°.
Roll dough to a 1/8-inch thickness on a lightly floured surface. Cut with a 2-inch round cookie cutter into 30 cookies, rerolling scraps as necessary. Place cookies 1 inch apart on baking sheets lined with parchment paper. Bake at 375° for 10 minutes or until edges of cookies are lightly browned. Remove from pans; cool completely on a wire rack.
To prepare frosting, combine powdered sugar, 1 1/2 tablespoons milk, and 1/4 teaspoon vanilla in a small bowl, stirring with a whisk until smooth. Spread 1 teaspoon frosting on each cookie; top with sprinkles. Place cookies on wire rack until set.
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