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Position oven racks in the top and bottom thirds of the oven and preheat to 375 degrees F. Line 2 baking sheets with parchment paper.
Whisk together the flour, baking powder and salt in a small bowl.
Beat the sugar and butter with an electric mixer on medium-high speed in a large bowl until light and fluffy, about 4 minutes. Add the vanilla and egg and beat to incorporate. Add the flour mixture in 2 batches and beat on medium-low speed until incorporated.
Put some sugar in a small bowl. Scoop heaping tablespoonfuls of the batter, roll them into balls and coat in the sugar. Arrange the balls on the prepared baking sheets about 2 inches apart. (There will be additional batter for a second batch.)
Bake until the bottoms and edges are barely golden brown, 12 to 14 minutes. Let the cookies cool for a few minutes on the baking sheets, then transfer them to a wire rack to cool completely. When the baking sheets are cool, repeat with the remaining batter. The cookies can be stored in an airtight container for up to 3 days.