In a large bowl using a hand mixer or in a stand mixer, beat the room temperature butter and sugar on medium speed until the mixture is light and fluffy, about 2-3 minutes.
Add the egg and vanilla extract. Mix on low speed until just combined, scraping down the sides of the bowl as needed.
Add the flour and salt to the bowl. Mix on low speed until a soft dough forms. Do not overmix.
Divide the dough into three equal portions. Leave one portion plain. Add a few drops of red food coloring to the second portion and green food coloring to the third. Mix each colored dough until the color is uniform.
Form each portion of dough into a flat disc, wrap tightly in plastic wrap, and refrigerate for 1 hour.
On a lightly floured sheet of parchment paper, roll each dough disc into a rectangle about 7x9 inches and ⅛-inch thick. Stack the three rectangles of dough. I put green on the bottom, white in the middle, and red on top.
Starting from a long edge, tightly roll the stacked dough into a log. Pour the sprinkles onto a plate or small baking sheet and roll the log in the sprinkles to coat the outside completely.
Wrap the dough log tightly in plastic wrap and refrigerate for at least 1 hour, or overnight. This chill is essential for clean slices.
Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper. Unwrap the chilled log and use a sharp knife to slice it into ¼-inch thick rounds. Place the cookies about 2 inches apart on the prepared baking sheet.
Bake for 10-12 minutes, or until the edges are set and the tops look dry. The colored dough will not show browning. Let the cookies cool on the baking sheet for a few minutes before moving them to a wire rack to cool completely.