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Preheat oven to 350°F. Line a baking sheet with parchment paper.
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Whisk together flour, baking powder, and salt.
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In a large bowl, beat the butter and sugars until light and fluffy. Mix in egg and vanilla.
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Add the dry ingredients to the butter mixture, mixing until combined.
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Roll the dough into balls. Flatten the balls slightly. Place on the baking sheet at least 2 inches apart.
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Bake the cookies for about 8-10 minutes. Let cookies rest on the baking sheet for 5 minutes and then transfer to a wire rack to cool completely.
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To make the frosting, mix the butter until creamy and smooth. Mix in powdered sugar and vanilla.
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Using a piping bag or a spoon, frost the bottoms of half the cookies, then sandwich with another cookie.