01 - Preheat oven to 350°F (175°C). 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 until evenly distributed. Set aside.
03 - In a large mixing bowl, beat the softened unsalted butter and granulated sugar using an electric mixer on medium speed until light and fluffy, approximately 2-3 minutes. Add the egg and vanilla extract, beating until fully incorporated.
04 - Gradually add the dry flour mixture to the creamed butter mixture, mixing on low speed until a smooth, cohesive dough forms. Avoid overmixing.
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 sombrero crowns.
06 - Arrange both the large and small cookie circles on the prepared baking sheets, leaving about 1 inch of space between each. Bake for 8-10 minutes, or until the edges are lightly golden. Transfer to a cooling rack and let cool completely.
07 - While the cookies cool, whisk together the confectioners' sugar, milk, and vanilla extract until smooth and pourable. Divide the icing among separate bowls and tint each with desired food coloring.
08 - Spread a layer of colored icing over each large cookie circle (the brim). Attach a small cookie circle (the crown) upright in the center using icing as adhesive. Top each crown with a gumdrop, securing with additional icing.
09 - Pipe or spread icing along the edge of each brim and apply colorful sprinkles. Allow the assembled cookies to set at room temperature until the icing firms up before serving.