In a large mixing bowl, whisk together flour, baking powder, and salt. Set aside.
Using a stand mixer or handheld mixer, beat butter and sugar together until light and fluffy, about 2 minutes.
Add eggs one at a time, mixing between each addition. Add vanilla extract and beat until evenly combined.
Add dry ingredients in three increments, mixing on low between each addition. Continue mixing on low until no dry spots remain (do not over mix).
Lightly dust a work surface with flour.
Working with about 1/3 of the dough at a time, roll it out to about 1/4-inch thick. Use cookie cutters to cut the dough into shapes and transfer them to a lightly greased sheet pan (you can also line it with parchment paper or a Silpat baking mat), leaving space between each cookie.
Transfer the sheet pan(s) with cookies to the freezer for 15 minutes. (This will minimize spreading as the cookies bake.)
While the cookies chill, preheat oven to 350°F.
When the oven is heated and the cookies are done chilling, transfer sheet pans with cookies directly to the oven and bake until the cookies are just set and show a hint of golden brown around the edges, 8 to 10 minutes (if cookies are frozen solid, they may take 3 to 5 minutes longer).
Transfer to a cooling rack and cool completely before adding frosting.