Mix together the flour, baking powder, and salt, by hand, in a medium mixing bowl.
In a large bowl, beat the butter and sugar with an electric mixer for about 3 minutes, until creamy.
Add the egg and vanilla and beat on medium speed until blended.
Add the flour mixture to the wet ingredients. Mix on low speed just until blended.
Divide the dough into half. Press each mound of dough into a flat disk. Wrap each disk with plastic wrap and refrigerate until well chilled, about 1 hour.
Preheat the oven to 350 degrees. Grease 3 baking sheets with butter.
Sprinkle your work surface with some flour. Unwrap 1 chilled disk of dough and place it on the floured surface. Sprinkle the top of the dough with a little more flour.
Roll out the dough with a rolling pin until it's about 1/4" thick. Sprinkle a bit more flour on the dough as needed so it doesn't stick.
After rolling out dough, cut out shapes with cookie cutters, and place the shapes on the greased cookie sheet.
Bake the cookies for 10-12 minutes, until light brown around the edges.
Let cool on the cookie sheet set on a baking rack for 15 minutes, and then transfer the cookies from the cookie sheet to the cooling rack and let them cool completely.