Cream sugar and butter with electric mixer; add egg. Beat in flour, baking powder, salt and vanilla. Form into two disc shapes, wrapped in plastic wrap. Chill 20 to 30 minutes.
Roll one portion of dough to 1/4-inch thickness on lightly floured surface. Cut with cookie cutters and place on ungreased cookie sheets. Repeat with remaining dough.
Bake in preheated 350°F oven 7 to 8 minutes or until edges are golden. Cool 1 to 2 minutes on baking sheet and then transfer to cooling rack. Ice and decorate as desired.