01 - Preheat oven to 350°F. Line two baking sheets with parchment paper and set aside.
02 - In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
03 - In a large mixing bowl, beat the softened butter and granulated sugar with an electric mixer until light and fluffy, approximately 2 to 3 minutes. Add the egg and vanilla extract, beating until fully incorporated.
04 - Gradually add the flour mixture to the wet ingredients, mixing on low speed until a smooth, cohesive dough forms. Do not overmix.
05 - Roll the dough out on a lightly floured surface to ¼-inch thickness. Using a 2-inch round cookie cutter, cut out 24 large circles for the sombrero brims. Using a 1-inch round cutter, cut out 24 small circles for the crowns. Re-roll scraps as needed.
06 - Arrange both large and small circles on the prepared baking sheets. Bake for 8 to 10 minutes, or until edges are lightly golden. Transfer to a cooling rack and let cool completely.
07 - While cookies cool, whisk together confectioners' sugar, 3 tablespoons of milk, and vanilla extract until smooth and pourable. Add additional milk as needed to reach desired consistency. Divide the icing among separate bowls and tint each with food coloring as desired.
08 - Spread colored icing over each large cookie circle to form the brim. Place a small cookie circle upright in the center using icing as adhesive. Attach a gumdrop to the top of each small cookie crown with a dab of icing. Decorate the brim edges with additional icing and colorful sprinkles. Allow icing to set completely before serving.