Knead the all-purpose flour into the sugar cookie dough on a lightly floured surface until smooth, about 5 minutes. Divide the dough into thirds.
Knead the red food coloring into one piece of dough, the blue food coloring into another piece and leave the third piece plain. Flatten and wrap each piece in plastic wrap, then refrigerate for about 20 minutes.
Roll each dough piece between 2 pieces of parchment to a roughly 11-by-7-by-1/8-inch rectangle. Wrap in plastic wrap and refrigerate again, about 20 minutes.
Stack the dough rectangles in red, white and blue order using the parchment to help you transfer them. Use the parchment to help you roll the dough stack into a tight spiral, forming an 11-inch-long log. Pour the sprinkles onto a baking sheet, then roll the log in the sprinkles so it is completely covered in them. Wrap the log in plastic wrap and freeze until solid, about 2 hours.
Position oven racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment.
Cut 1/4 inch off each end of the log with a very sharp knife to reveal the swirly pattern. Cut the log into slices about 1/4 inch thick. Arrange the slices about 1 1/2 inches apart on the prepared baking sheets. Bake, rotating the baking sheets halfway through, until the cookies are crisp and just starting to turn brown on the bottom, 10 to 12 minutes. Let cool on the baking sheets for 10 minutes before transferring them to a cooling rack to cool completely.
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