Using electric mixer, beat sugar and butter in large bowl until light. Beat in egg and vanilla. Add flour, baking powder and salt. Mix just to combine. Divide dough in half. Gather each piece into ball; flatten into disks. Wrap in plastic and chill 1 hour.
Preheat oven to 325°F. Roll out 1 dough disk on floured work surface to 1/8-inch thickness. Using assorted cookie cutters, cut out cookies. Transfer cookies to ungreased baking sheets, spacing 1 inch apart. Gather dough scraps; re-roll to 1/8-inch thickness. Cut out more cookies.
Bake cookies until pale golden, about 13 minutes. Transfer cookies to racks; cool. Repeat with remaining dough disk.
Spoon melted chocolate into pastry bag fitted with 1/16-inch plain tip. Pipe chocolate onto cookies. Let cookies stand until chocolate sets. (Can be made ahead. Store in airtight container at room temperature up to 1 week, or freeze up to 1 month.)