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Beat the butter until creamy. Scrape down the sides and bottom of the bowl. Beat in the confectioners' sugar until light and fluffy. Scrape down the sides and bottom of the bowl again.
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Beat in the egg yolks, one at a time. Beat in the peppermint extract. Scrape the bowl.
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Whisk the salt into the flour in a separate bowl.
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Slowly mix the flour into the dough, one third at a time. Finish the mixing by hand and be certain to not over-mix.
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Wrap the dough in plastic wrap and chill for 30 minutes.
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Divide the dough into 2 equal pieces. Roll each piece into a log, and then roll the log into plastic wrap. Tightly seal each log and then place into the fridge to chill for at least 2 hours, preferably overnight.
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When you are ready to bake, preheat the oven to 350 F.
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Slice the cookies 1/3-inch thick and place on a cookie sheet lined with parchment paper or a silicone mat. The cookies should be about 1/2-inch apart--I got all cookies baked onto 2 cookie sheets.
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Sprinkle the cookies with the coarse sanding sugar.
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Bake for 12-14 minutes, until cookies are set and just barely starting to brown. Let cool completely on the cookie sheets.
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Serve with ice cream or gelato of choice--I liked coffee, mint and dark chocolate best.