In a large bowl using a handheld or stand mixer fitted with the paddle attachment, beat 3/4 cup butter until creamed and smooth – about 1 minute. Add 1/3 cup sugar and beat on high speed until light and fluffy, about 3 or 4 minutes. Scrape down the sides and bottom of the bowl as needed.
Add 1 egg, 2 tsp vanilla extract and beat on high until fully combined, about 2 minutes. Scrape down the sides and bottom of the bowl as needed.
In a separate bowl, whisk the flour, baking powder, and salt together. Turn the mixer down to low speed, and add about half of the flour mixture, beating until just barely combined. Add the rest of the flour and continue mixing until just combined. If the dough still seems too soft, you can add 1 tbsp more flour to get to a better consistency for rolling.
Take the dough out onto a lightly floured clean counter. Divide the dough into 2 equal parts. Roll each portion out onto a piece of parchment to about 1/4″ thickness. Stack the pieces (with paper) onto a baking sheet, cover lightly, and refrigerate for at least 1-2 hours and up to 2 days. Chilling is very important and will effect the final product. Don't skip this part!
Once chilled, preheat the oven to 350°F (177°C). Line 2-3 large baking sheets with parchment paper or a silicone baking mat (you can use the same parchment from when you chilled the dough). The number of baking sheets you need will depend on how large/small you cut your cookies. Remove one of the dough pieces from the refrigerator and using a cookie cutter, cut into desired shapes. Transfer the cut cookie dough to the prepared baking sheet. Re-roll the remaining dough and continue cutting until all is used up. The key is to keep the dough cold until right before you bake.
Bake for 10-12 minutes, until very lightly starting to get golden colored on top and around the edges. Make sure you rotate the baking sheet halfway through bake time. Allow to cool on baking sheet for 5 minutes, then transfer to a wire rack to cool completely before icing. No need to cover the cookies as the cookies cool.