In a medium bowl, sift together the flour, baking powder and salt.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter until smooth, about 2 minutes. Add the sugar and continue beating until the mixture is light and fluffy, about 3 minutes. Add the eggs one at a time, beating between each addition, then add the vanilla.
Turn the mixer off. Add the flour and then beat just until combined.
Remove the dough, shape it into a disk about 6 inches wide and wrap it securely in plastic wrap. Refrigerate the dough for a minimum of 1 hour. (See Notes.)
When ready to bake, preheat the oven to 350ºF and line two baking sheets with parchment paper.
Remove the dough from the fridge and cut it in half. Roll half the dough between two pieces of wax paper until it's about 1/4-inch-thick. Using cookie cutters, cut out your desired shapes and place them roughly 2 inches apart on the cookie sheets. Repeat the rolling and cutting process with the remaining dough. (Leftover scraps can be re-rolled.)
Refrigerate the cut-out cookies for 15 minutes. (This helps them keep their shape when baking.)
Bake the cookies for 9 to 11 minutes until pale golden, and then transfer them to a rack to cool completely. Frost and decorate as desired.
The dough can be refrigerated overnight, however when you remove it from the fridge, you will likely need to let it rest at room temperature for some time before it's soft enough to roll.
When in doubt, remove the cookies from the oven just prior to them turning pale golden. The cookies will crisp slightly as they cool, so removing them while slightly underbaked guarantees your cookies will remain soft. For crispier sugar cookies, bake them the full time.