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1. Place egg in small saucepan, cover with 1 inch water, and bring to boil over high heat. Remove pan from heat, cover, and let sit 10 minutes. Meanwhile, fill small bowl with ice water. Using slotted spoon, transfer egg to ice water and let stand 5 minutes. Crack egg and peel shell. Separate yolk from white; discard white. Press yolk through fine-mesh strainer into small bowl.
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2. In bowl of stand mixer fitted with paddle attachment, beat butter, granulated sugar, salt, and cooked egg yolk on medium speed until light and fluffy, about 4 minutes, scraping down sides of bowl and beater with rubber spatula as needed. Turn mixer to low, add vanilla, and mix until incorporated. Stop mixer; add flour and mix on low speed until just combined, about 30 seconds. Using rubber spatula, press dough into cohesive mass.
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3. Divide dough in half; roll each piece into log about 6 inches long and 1 3/4 inches in diameter. Wrap each log in 12-inch square of parchment paper and twist ends to seal and firmly compact dough into tight cylinder (see illustrations below). Chill until firm, about 1 hour.
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4. Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Line 2 rimmed baking sheets with parchment paper. Using chef's knife, slice dough into 1/4-inch-thick rounds, rotating dough so that it won't become misshapen from weight of knife. Place cookies 1 inch apart on baking sheets. Using pastry brush, gently brush cookies with egg white mixture and sprinkle evenly with turbinado sugar.
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5. Bake until centers of cookies are pale golden brown with edges slightly darker than centers, about 15 minutes, rotating baking sheets front to back and top to bottom halfway through baking. Cool cookies on baking sheet 5 minutes; using thin metal spatula, transfer cookies to wire rack and cool to room temperature. Store cooled cookies between sheets of parchment paper in airtight container for up to 1 week.
