1Whisk together the flour, baking powder, and salt in a medium bowl to break up any lumps; set aside.
2Place the butter and sugar in the bowl of a stand mixer fitted with a paddle attachment and beat on medium speed until lightened in color and fluffy, about 3 minutes. Stop the mixer and scrape down the sides of the bowl and the paddle with a rubber spatula.
3Return the mixer to medium speed, add the egg and vanilla, and beat until incorporated. Stop the mixer and scrape down the sides of the bowl and the paddle with the rubber spatula.
4Turn the mixer to low speed and slowly add in the reserved flour mixture. Mix until just incorporated.
5Turn the dough out onto a clean work surface and divide it in half. Roll each portion into a log about 1 1/2 inches in diameter, wrap in plastic wrap, and refrigerate until firm, at least 2 hours and up to 3 days.
6When ready to bake the cookies, heat the oven to 350°F and arrange the racks to divide the oven into thirds.
7Remove the dough logs from the refrigerator, remove the plastic wrap, and slice into 1/4-inch rounds. Place the rounds on 2 baking sheets about 1/2 inch apart (about 20 cookies will fit on each sheet).
8Place both sheets in the oven and bake for 6 minutes. Rotate the baking sheets front to back and top to bottom and bake until the edges of the cookies are firm but the tops are still soft, about 6 to 7 minutes more.
9Place the baking sheets on wire racks and let cool for 5 minutes. Using a flat spatula, transfer the cookies to the wire racks to cool completely.