Remove half of the cookie dough to a separate bowl. Colour one half with red gel food colouring and mix until you reach the desired colour. It may take more food colouring than you think.
Place each dough ball on a separate large sheet of parchment paper. Sprinkle with gluten-free flour on the top and bottom of the dough. Roll each dough into a rectangle as large as you can. They will be quite thin. You will trim the edges later. Place the parchment-lined doughs in the fridge for 20 minutes to firm up slightly. Do not leave for longer because they will be too firm to roll. (If you're using dairy-free butter, they may take longer to firm up).
Remove from the fridge and trim the edges into a clean rectangle. My rectangles were about 9.5x14". It's okay if yours are larger or smaller, as long as the red and white doughs are even. Combine the scraps for an extra cookie or two!
Use the parchment paper to help you place the red dough on top of the white dough, dough sides facing each other. Press down firmly to ensure they stick together, and remove the top parchment paper. From the long side, use the bottom parchment paper to help you roll the dough into a spiral. Roll tight and firmly to keep its shape. If the dough is sticking, put it back in the fridge for 10 minutes and try again.
Pour the sprinkles into a large rimmed baking sheet. Place the log of dough on the baking sheet and roll in the sprinkles to cover the outside. Wrap the log in plastic wrap or parchment paper and place in the fridge for at least 2 hours, ideally overnight.