In a medium mixing bowl, stir the flour, baking powder, and salt together. Set aside.
In a large mixing bowl, beat the butter and sugar for 1-2 minutes, until pale and fluffy. Add the egg and vanilla and beat until combined. Add the dry ingredients to the wet ingredients in two additions, mixing until just combined after each addition. Scrape the sides of the bowl as needed.
Divide the dough into 3 equal portions. (You can eyeball it or use a kitchen scale to measure the pieces.) Place ⅓ of the cookie dough back in the mixing bowl and add the red food coloring. Beat with your mixer until the color is completely incorporated. (Be sure to use a bowl that won't stain.) Clean the hand mixer blades (or stand mixer paddle) and wipe out the bowl. Place another ⅓ of the cookie dough into the bowl and add the green food coloring. Beat until the color is completely incorporated. Leave the third piece of dough as-is. You should now have red, green, and "white" cookie dough.
Press each piece of dough into a flat square and wrap tightly in plastic wrap. Refrigerate the dough for 1 hour or until firm.
Remove the cookie dough squares from the fridge. Place the red dough between 2 sheets of parchment paper or waxed paper. Use a rolling pin to roll the dough out into a 10-inch square. Leave the dough between the parchment paper pieces. Repeat with the red and white cookie dough. Refrigerate for 30 minutes.
Remove dough from the fridge, and take the top layer of parchment paper off each sheet of cookie dough. Place the red dough on a work surface. Top with the white dough, non-paper sides touching, being sure to try and line up the edges of the dough as much as possible. Remove the other piece of parchment from the white dough. Place the green dough over the white dough, non-paper sides touching, while lining up the edges. Remove the parchment from the green dough.
Use a pizza cutter or knife to trim the left and right edges of the cookie dough into straight lines. Very carefully, starting with one of the straightened sides, roll the dough into a log shape until you reach the other trimmed edge. Use the parchment paper to lift the dough and get things started. Once rolled, gently pinch the seam closed, along with any cracks in the outer red layer. Wrap the dough log in plastic wrap and refrigerate for 1 hour or until firm.
Preheat your oven to 350 degrees F. Line a baking sheet with a silicone mat or parchment paper.
Use a sharp knife to cut the dough log into ¼-inch thick slices, discarding the ends since they don't make a nice pinwheel design. (Or bake them off as a chef's snack!) Place the cookies on the prepared baking sheet and bake for 12 to 15 minutes, or until the cookies are set and the bottoms are barely golden.
Carefully remove from the oven and let cool for 5 minutes on the baking sheet before transferring to a wire rack to cool completely. Serve immediately or store in an airtight container until ready to serve.
Video
Notes
I wanted very vibrant colors, so I used a lot of food coloring. You can start with 1/4 teaspoon for each color and add 1/4 teaspoon more at a time until you reach your desired color. If you are using gel food coloring, you will not need nearly as much. Start with a few drops, mix, and then add more color as needed.
Once you layer and roll the cookie dough, it can be kept in the fridge for up to 3 days or frozen for up to 3 months. If freezing, wrap in plastic wrap and then place in a food storage bag. Thaw overnight in the fridge before slicing.