Whisk together flour, baking powder, and salt. In a large bowl, beat butter and sugar until light, 2-3 minutes. Beat in egg and vanilla, then mix in dry ingredients just until combined.

Divide dough into three equal portions. Knead red gel coloring into one and blue into another until even; leave the third plain.

Between parchment, roll each portion into a 9x6-inch rectangle, 1/4 inch thick. Stack red, white, then blue; gently press layers. Starting from a long edge, roll up tightly into a log, using the parchment to help. Pinch seam to seal.

Wrap log in plastic; chill at least 2 hours until very firm. Slice into 1/4-inch rounds with a sharp knife.

Preheat oven to 350°F. Place rounds on parchment-lined sheets, 2 inches apart. Bake 8-10 minutes until edges are set. Cool on sheet 5 minutes, then transfer to wire rack.