Cream together butter and granulated sugar until fluffy.
Mix in the egg and vanilla until pale and fluffy.
Add flour, salt, and baking powder, and mix until combined.
Chill the dough for 1 hour, then roll it out on a floured surface.
Cut out cookies using cookie cutters, then chill them for at least 1 hour.
Preheat the oven to 350 degrees, then bake the cookies for 10-12 minutes.
Let them cool completely before icing.