Add butter and granulated sugar to a large bowl. Cream together until fluffy, 1-2 minutes.
Add in the egg and vanilla and mix until pale in color and fluffy, 1-2 minutes.
Then add in the flour, salt and baking powder. Mix until combined.
Form the dough into a disk and wrap it tightly in plastic wrap. Chill the dough in the fridge for 1 hour.
Line a baking sheet with parchment paper.
On a well floured surface roll the dough out until it's ¼ inch thick. (If the dough is too cold and difficult to work with, let it sit for 10-20 minutes then roll it out.)
Cut out cookies using cookie cutters.
Transfer the cut cookies to the parchment paper lined baking sheet. Chill the cut cookies for at least 1 hour, or until ready to bake. (You can chill the cookies together on one or two baking sheets, and then separate them out when ready to bake.)
When ready to bake the cookies, preheat the oven to 350 degrees. (Line another baking sheet with parchment paper if the cookies are together on one sheet.)
Arrange the chilled cookie cutouts about 2 inches apart from each other.
Bake regular sized cookies, that are roughly 3 ½ inches for 10-12 minutes. Bake mini cookies, that are roughly 2 inches for 6-8 minutes. Watch them closely to make sure the edges don't get too browned. They should be a light golden color.