In a medium bowl, stir together the flour, baking powder and salt, and set aside.
In the bowl of an electric mixer, using the paddle attachment, cream together the butter and sugar until light and fluffy. This should take 2-3 minutes.
Add the egg and vanilla and beat another minute or so.
Add the flour mixture slowly. (Be careful not to add too much at a time or you will have a snowstorm of flour.)
Blend until all of the flour is incorporated. The mixture will look crumbly at first but just keep blending until the dough begins to pull away from the sides of the bowl.
Remove about half of the dough from the bowl and knead by hand just a bit to form a smooth ball. Roll out dough onto a lightly floured work surface, or onto a piece of plastic wrap.
Cut out circles using your larger circle cutter. For each finished cookie box you'll need 4 circle cookies. I used a 2.5" round cookie cutter, but feel free to adjust the size as you wish. Place entire baking sheet of unbaked cookies into the freezer for 5 minutes. Preheat oven to 350°F and bake cookies in preheated oven for 9 to 11 minutes, depending on the size and thickness of your cookies. Bake the cookies WHOLE (do not cut out the centers until after the cookies are baked).
The cookies are baked when then are no longer shinny on top, but they will not show much of any change in color.
As soon as the cookies are baked and you have removed them from the oven (but they are still hot) use the smaller circle to cut into the center of HALF of your cookies. Cut through the dough with the smaller cutter, but do not touch the cookies (leave the cutout center in place). Allow the cookies to cool a minute or two on the baking sheet, then use a spatula to carefully move the cookies, with the cutout intact, onto the cooling rack.
Once the cookies have cooled fully, THEN remove the inner circle. You may need to use the smaller cutter to make sure the inner circle is cut cleanly through. Once you have each of the piece you need, set them together in sets with one whole cookies as the bottom of the "cookie box", then add two rings and then you'll need one whole cookie as the top (see post for details).