1. Into a medium bowl, whisk together flour and baking powder. Set aside.
2. In a large bowl, using an electric mixer fitted with the paddle attachment on medium speed, beat butter and sugar for about 1½ minutes, or until smooth. Beat in the egg and vanilla until well combined. Scrape the sides of the bowl with a rubber spatula.
3. Turn the mixer speed to low and carefully add the flour mixture a little at a time, occasionally stopping to scrape the sides of the bowl. Once all the flour has been incorporated, the dough should form a ball around the mixing attachment and feel soft but not sticky.
4. Wrap the dough ball in a piece of plastic wrap press down to create a disc shape. Refrigerate for at least 10 minutes before rolling.
5. Preheat the oven to 375°F.
6. Roll out the dough on a lightly floured flat work surface to about ¼-inch thickness, using additional flour as necessary to prevent sticking. Use cookie cutters to create desired shapes, and carefully transfer with a cookie spatula to a nonstick baking sheet, placing the cookies about ¾ inch apart.
8. Bake one sheet at a time in the middle of the oven until puffy, about 7-8 minutes. Allow the cookies to rest for 2 minutes on the baking sheet before transferring to a wire rack to cool completely.
9. Repeat with the gathered scraps and remaining disk until all the dough has been used.