Step 1In a large bowl, combine the butter, sugar, lime zest, vanilla and salt and beat with an electric mixer on low speed until smooth, about 1 minute (see Baker’s Note). Mix in the peppermints, then add the flour and mix on low speed until just combined.
Step 2Split the dough in half and place each half on a sheet of parchment paper, forming each into a rough log lengthwise down the center of the paper. Fold one long edge of the paper over the log to cover it, then use a ruler or bench scraper to press the paper against the dough to tighten it into an even log, roughly 12 inches long. Wrap the log up completely in the paper and twist the ends like candy wrappers to close. Repeat with the other dough log, then refrigerate both logs until firm, at least 30 minutes.
Step 3Unwrap 1 dough log and brush it all over lightly with some beaten egg white. Sprinkle the log with ½ cup red sparkling sugar until completely coated. Transfer the log to a sheet of plastic wrap and wrap up completely. Return to the refrigerator to allow the egg white and sugar to set, at least 30 minutes or up to overnight. Repeat with the second dough log, more egg white and the remaining ½ cup sparkling sugar.
Step 4Heat the oven to 350 degrees. Line 2 baking sheets with parchment paper.
Step 5Unwrap 1 log and cut into ¾-inch-thick disks, rotating the log a quarter turn after each slice to keep each disk round (you should get at least 15 cookies, minus the trimmed ends). Arrange the disks evenly on one of the prepared baking sheets. Repeat with the second log and second baking sheet.
Step 6Bake, rotating the baking sheets from front to back and top to bottom halfway through, until cookies are set and the edges are very lightly browned, about 18 minutes.
Step 7Let the cookies cool on the baking sheets for 1 minute before transferring to a wire rack to cool completely.