Make the cookies: Sift flour, baking powder, and 1/2 teaspoon salt into a large bowl.
Beat butter and sugar with a mixer on medium-high speed until pale and fluffy, about 3 minutes. Beat in eggs, 1 at a time. Reduce speed to low. Add flour mixture, then vanilla. Refrigerate dough, wrapped in plastic wrap, for at least 1 hour.
Preheat oven to 325 degrees. Roll out dough to a scant 1/4-inch thickness on a floured surface. Cut out cookies using a 1 3/4-, 2 1/4-, 2 3/4-, or 3 1/2-inch round cookie cutter, rerolling scraps once. Transfer to a baking sheet. Refrigerate until firm, about 30 minutes.
Bake until edges just start to brown, 17 to 19 minutes. Transfer cookies to a wire rack, and let cool completely.
Decorate the cookies: Pipe an outline of white icing around edge of 1 cookie, leaving a 1/4-inch border, then "flood" with more white icing to cover.
Immediately pipe a red or blue dot in the center of cookie. Then pipe concentric rings of colors around the center dot (using the same color as the dot, or alternating colors).
Immediately drag a toothpick through the colors to create bursts, starting from the center dot and working toward the edge, then alternate dragging inward and outward as you work around the cookie. (Or drag around the cookie in 1 direction or curve the lines for a pinwheel effect.) Let dry. Repeat with remaining cookies and icings.
Cookie dough can be refrigerated for up to 5 days or frozen for up to 1 month. Thaw before using. Storage: Decorated cookies can be stored in an airtight container at room temperature for up to 3 days.