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To make the cookie dough: In the bowl of a stand mixer fitted with the flat beater attachment or working in a large bowl with an electric hand mixer, beat the butter, sugar, and salt until light and fluffy, 2 to 3 minutes at medium speed.
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Add the egg yolk, vanilla, and peppermint extract and beat until incorporated, stopping to scrape down the sides of the bowl as needed.
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Add the flour and mix on low until a crumbly dough forms.
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Use a scale to divide the dough into three portions: 470g (this will become the white dough), 157g (this will become the red dough), and 78g (this will become the green dough). If you don’t have a scale, divide the dough into thirds and then combine two of the thirds to form the larger white dough; it should be about 1 ½ cups of dough. Divide the remaining dough into thirds, then combine two thirds to form the medium size piece of dough. This will be the red dough. The remaining one third of dough will be the green dough. Transfer each piece to a bowl, cover the white dough (largest portion), and set aside.
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To make the red dough: In a very small microwave-safe bowl, stir together the packet of red food coloring and water. Microwave in 2-second intervals, stirring between each, until the food coloring dissolves and the mixture is smooth.
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Add all of the red food coloring mixture to the bowl with the larger amount of dough (157g). Stir together with a flexible spatula until combined.
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To make the green dough: In a very small microwave-safe bowl, stir together the packet of green food coloring and water. Microwave in 2-second intervals, stirring between each, until the food coloring dissolves and the mixture is smooth.
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Add all of the green food coloring mixture to the bowl with the smaller amount of dough (78g). Stir together with a flexible spatula until combined.
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To assemble: Working on a piece of parchment or on a clean, lightly floured surface, divide the white dough into six pieces, about 78g each. Round each piece into a ball, then roll into logs about 8" long. Set aside.
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On a floured surface, divide the red dough into three pieces, about 54g each. Round each piece into a ball, then roll into logs, about 8" long. Set aside.
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On a floured surface, divide the green dough into two pieces, about 42g each. Round each piece into a ball, then roll into logs, about 8" long. Set aside.
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Use your hands to slightly flatten one log of white dough to 1 1/2" to 2" wide. Place a red log on top of the flat log, then use your fingers to slightly flatten the red log to the same width, about 1 1/2".
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Repeat with the remaining logs of dough and stack as follows; white, green, white, red, white, green, white, red, white — until you have an 11-layered stack of flattened logs.
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Use your hands to gently roll the stack of flattened logs of dough to round the edges and form one large log about 8" long. Twist the ends of the large log in opposite directions to create a swirled effect.
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To make a perfectly round log, place the log of dough on a sheet of parchment, then fold the parchment toward you to encase the dough. Use your nondominant hand to press a straight object (a ruler or a bench knife works well) between the dough and the work surface, then use your dominant hand to gently tug the parchment toward you. This should pull the dough so that it butts up against the straight object and makes a very smooth, symmetrical log. See this blog post, How to make round cookies, for a visual demonstration.
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To finish: Spread the sparkling sugar on a piece of parchment in a 2"-wide strip about 8" long. Roll the dough log in the sugar to coat.
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Wrap the sugar-coated log in plastic wrap or transfer to a Large Slice & Bake Dough Keeper. Cover and refrigerate until very firm, at least 2 hours and up to 4 days.
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When ready to bake, preheat the oven to 350°F with a rack in the upper and lower third. Line two baking sheets with parchment.
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Remove the dough from the refrigerator and unwrap or remove the dough from the Keeper. Using a sharp knife, slice the dough into 1/4"-thick rounds until you have 24 cookies; rotate the log about 90° (a quarter turn) every three to four slices to maintain a neat circular shape if needed. Transfer the slices to the prepared baking sheets, 12 cookies per pan, spacing about 2" apart. Re-wrap the remaining dough or place it in the Keeper then return to the refrigerator.
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Bake the cookies for 12 to 15 minutes, rotating the pans from top to bottom and front to back halfway through baking, until they’re golden brown all over with slightly darker edges.
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Remove from the oven and let the baking sheets cool on a wire rack. When the first batch is completely cool, repeat the slicing process with the final piece of dough and bake.
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Storage information: Store leftover peppermint cookies, well-wrapped, at room temperature for up to 1 week; freeze for longer storage.
