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Prepare two baking sheets with parchment paper and set aside.
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In small bowl, combine flour, baking powder and salt.
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In large bowl, mix butter and sugar until combined. Add egg and vanilla and mix thoroughly. Add dry ingredients and mix until just combined.
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Shape dough into ball, wrap with plastic and chill 1 hour.
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Preheat oven to 325 degrees F.
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Place chilled dough between two pieces of parchment paper and roll to 1/8-inch thickness, removing and replacing parchment paper as needed.
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Using lightly floured 2-inch round cookie cutter, cut cookies and place onto prepared baking sheets about 1 inch apart.
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Bake 15-18 minutes, rotating halfway through baking, until cookie edges are slightly golden.
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Let cook on baking sheets 10 minutes, then remove to wire racks to cool completely.
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To make frosting, cream butter until fluffy. Add vanilla and salt.
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Gradually add 2 cups powdered sugar then milk 1 Tbsp. at a time.
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Add desired food coloring and frost cookies.