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 two equal pieces. Shape one piece of the dough into an 8-inch square, wrap it securely in plastic wrap and place it in the fridge. (This will be the white portion of the cookies.)
Return the remaining piece of dough to the stand mixer bowl, and with the mixer on "low," add in the red food coloring, drop by drop, until it reaches your desired pink color. Remove the pink dough, shape it into an 8-inch square, wrap it securely in plastic wrap and place it in the fridge.
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 white dough in between two pieces of wax paper and roll it into a rectangle about 1/8-inch thick. Roll out the pink 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 white dough and then peel the wax paper off one side of the pink dough and use the other side to transfer the pink dough on top of the white dough. Very lightly roll the two 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. Unwrap the dough logs and place the sprinkles in a large, shallow dish. Roll the dough logs in the sprinkles until they are completely coated. Re-wrap the dough logs in wax paper and plastic and refrigerate them 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 rack to cool completely.