Whisk together flour, baking powder, and salt in a small bowl.
Beat together butter and sugar in a large bowl with an electric mixer at medium-high speed until pale and fluffy, about 3 minutes in a standing mixer (preferably fitted with paddle attachment) or 6 with a handheld. Beat in egg and vanilla. Reduce speed to low, then add flour mixture and mix until just combined.
Form dough into a 12-inch log (2 inches in diameter) on a sheet of plastic wrap and roll up dough in plastic wrap. Chill dough on a baking sheet until firm, at least 4 hours. (To roll cookies into balls, see cook's note, below.)
Put oven racks in upper and lower thirds of oven and preheat oven to 375 degrees F.
Cut enough 1/8 to 1/4-inch-thick slices from log with a heavy knife to fill 2 ungreased large baking sheets, arranging slices about 1-inch apart (chill remainder of log, wrapped in plastic wrap). If garnishing with coarse sugar, sprinkle slices with it.
Bake cookies, switching position of sheets halfway through baking, until edges are golden, 12 to 15 minutes total. Cool on sheets 3 minutes, then transfer with a metal spatula to racks to cool completely. Make more cookies with remaining dough on cooled baking sheets.