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In the bowl of an electric mixer fitted with the paddle attachment, beat the butter, salt, and vanilla on high speed until fluffy, 2 to 3 minutes.
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Separate the eggs, placing the whites and yolks in separate bowls. Add the sugar to the yolks and whisk by hand until fluffy, about 1 minute. Add the yolk mixture to the butter mixture and beat on low to combine. Sift the flour and baking powder together, then add to the butter mixture and mix on low speed until the flour is incorporated. Do not overmix.
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Roll the dough into 3 logs, each 1-1/2 inches in diameter. Wrap in plastic wrap and refrigerate until firm, at least 1 hour.
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Position a rack in the middle of the oven and preheat to 350°F. Line baking sheets with parchment paper. Slice the dough logs into 40 (1/4-inch) rounds, place the rounds on the prepared baking sheets about 1 inch apart. Whisk the egg whites, then brush the cookie tops with them. Bake for 12 minutes, rotating halfway through. Let cool for 10 minutes on the baking sheet, then transfer to a wire rack to cool completely.