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.
Cook's Note: Sanding sugar is available at specialty foods shops and New York Cake & Baking Distributor (800-942-2639).
These cookies are absolutely delicious made with regular butter, but they become downright amazing with Plugra or Land O Lakes Ultra Creamy. Dough log can be chilled up to 5 days or frozen, wrapped in a double layer of plastic wrap, 1 month. If frozen, thaw dough in refrigerator just until it can be sliced. Instead of forming dough into a log and chilling, you can roll tablespoons of dough into 1 1/4-inch balls, then roll balls in finely chopped nuts (about 1 cup) and coarse sugar. Bake, switching position of sheets halfway through baking, until bottoms are browned, about 15 minutes total. To garnish cookies with chocolate, melt 3 1/2 ounces chocolate and cool slightly. Transfer to a heavy-duty sealable plastic bag and snip a 1/16-inch opening in 1 corner. Pipe chocolate evenly back and forth over cookies. Let chocolate set before storing cookies. Cookies keep, layered between sheets of waxed paper or parchment, in an airtight container at room temperature 1 week.