Combine flour, baking powder, and salt in medium bowl and whisk until combined. Set aside.
Combine butter and sugar in bowl of electric mixer and beat until smooth. Beat in eggs and vanilla. Add flour mixture in three parts. Once dough comes together, cover bowl with plastic wrap and refrigerate for at least 1 hour.
Preheat oven to 400ยบ F.
On a floured surace, roll out dough to 1/4 inch thickness. Using round cookie cutter or an overturned cup, cut dough into circles about 3 1/4 inches in diameter. Place cookies 1 inch apart on ungreased cookie sheet.
Bake 8 to 10 minutes, or until golden brown. Cool completely before icing.