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In the bowl of a stand mixer, combine butter and powdered sugar until fluffy.
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Add egg, vanilla and peppermint extract, scraping down sides of bowl as needed.
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Reduce mixer speed and add flour and salt, mixing until well combined.
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Divide dough in half.
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In the bowl of the stand mixer, combine 1/2 the cookie dough with red food coloring.
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Shape cookie dough into disc and cover tightly with plastic wrap.
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Refrigerate one hour or up to overnight.
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Once dough has chilled, preheat oven to 375 degrees.
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Measure one tablespoon of each color dough and gently roll into 6 inch long log shapes.
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Twist cookie dough together and transfer to parchment paper or silpat lined baking sheet.
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Chill shaped cookie dough for 15 minutes before baking.
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Bake 10-12 minutes, or until bottoms of cookies are a very light golden brown.
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While the cookies are baking, prepare peppermint glaze. Spoon over cookies after they have cooled slightly.