Preheat the oven to 350 degrees. Lightly oil 2 baking sheets.
In the bowl of your electric mixer, combine butter and sugar and beat until fluffy. Add the eggs and vanilla, beat until smooth.
Whisk together the flours and baking powder. Add to the creamed mixture, mixing on low speed until just combined. Divide the dough in half and press each piece into a disk. On a well-floured surface, roll the dough out to 1/8 inch thickness. Cut out cookies, rerolling the scraps of dough.
Bake for 5 to 7 minutes, until just golden on the edges. Let cool completely on a rack.