In a bowl, use a hand mixer on medium speed to beat together butter and sugar until pale and fluffy, 2–3 minutes. Add egg and beat until smooth. Add flour and mix on low speed until just combined. Transfer dough to a work surface and form into a ball; halve ball and form each half into a flat disk. Wrap disks separately in plastic wrap and refrigerate until chilled, 1 hour.
Heat oven to 350º. Transfer one disk to a lightly floured work surface; using a rolling pin, roll out to 1⁄4″ thickness. Using a 1 1⁄2″ round or fluted cookie cutter, cut out rounds and transfer to parchment paper-lined baking sheets, spacing them 1″ apart; repeat with remaining dough disk. Bake, rotating baking sheets top to bottom and front to back halfway through cooking, until cookies are set but not browned, 8–10 minutes. Let cool before serving.