Beat butter and sugar with an electric mixer in a large bowl until smooth. Blend egg and vanilla extract into the butter mixture.

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Stir flour, baking powder, and salt together in a bowl; mix into the butter mixture until just incorporated. Wrap dough in plastic wrap and refrigerate at least 1 hour. Remove from refrigerator and let the dough warm to room temperature.

Preheat oven to 350 degrees F (175 degrees C).

Roll dough out onto a flat surface to about 1/4 inch in thickness. Cut dough into shapes with cookie cutters; arrange onto baking sheets.

Bake cookies until firm in the center, but not yet browning, 8 to 12 minutes. Let cool briefly on baking sheet before transferring to a cooling rack to cool completely.