In a medium bowl, sift together the flour, baking powder and salt.

In the bowl of a stand mixer fitted with the paddle attachment, beat the butter until smooth, about 2 minutes. Add the sugar and continue beating until the mixture is light and fluffy, about 3 minutes. Add the eggs one at a time, beating between each addition, then add the vanilla.

Turn the mixer off. Add the flour and then beat just until combined. Remove the dough and separate it into three equal pieces. Shape one piece of the dough into a 4- by 4-inch square, wrap it securely in plastic wrap and place it in the fridge. (This will be the white portion of the cookies.)

Return one of the remaining pieces of dough to the stand mixer bowl, and with the mixer on "low," add in the red food coloring until it reaches your desired color. Remove the red dough, shape it into a 4- by 4-inch square, wrap it securely in plastic wrap and place it in the fridge. (This will be the red portion of the cookies). Clean out the bowl to remove any red food coloring residue.

Add the final piece of dough to the stand mixer bowl, and with the mixer on "low," add in the blue food coloring until it reaches your desired color. Remove the blue dough, shape it into a 4- by 4-inch square, wrap it securely in plastic wrap and place it in the fridge. (This will be the blue portion of the cookies.)

Refrigerate the dough for 30 minutes.

Remove the dough from the fridge and cut each square in half to form two rectangles. Wrap half of each color of dough in plastic wrap and return it to the fridge. Place the red dough in between two pieces of wax paper and roll it into a rectangle about 1/8-inch thick. Roll out the white dough between two separate pieces of wax paper until it is a rectangle about 1/8-inch thick. Roll out the blue dough between two separate pieces of wax paper until it is a rectangle about 1/8-inch thick.

Peel the top layer of wax paper off of the red dough and then peel the wax paper off one side of the white dough and use the other side to transfer the white dough on top of the red dough. Peel off the wax paper from one side of the blue dough and use the other side to transfer it atop the white dough. Very lightly roll the three layers together.

Starting at the shorter end of the rectangle, roll the dough as tightly as possible into a log. Repeat the rolling and stacking process with the remaining dough in the fridge. Wrap the logs in wax paper and then plastic wrap and refrigerate them for 1 hour.

Remove the dough logs and roll them on the counter several times so they don't develop a flat side. Alternately, stand the rolls up in glasses in the fridge. Refrigerate the dough for 4 more hours.

When ready to bake, preheat the oven to 350ºF and line two baking sheets with parchment paper. Remove the dough from the fridge and slice each log into 1/4-inch rounds. Place the rounds about 2 inches apart on the baking sheets, as the cookies will expand when baked.

Bake the cookies for 9 to 11 minutes until pale golden, and then transfer them to a wrack to cool completely.