Preheat the oven to 325° F. Line a baking sheet with parchment paper. Set aside.
Sift flour with baking powder and salt into a separate bowl. Set aside.
In the bowl of your mixer fitted with a paddle attachment, (or using a handheld beater) beat the butter, sugar, lemon zest, lemon juice and vanilla on low speed to blend. Increase speed and Beat until fluffy and well-blended.
Slowly add the flour mixture to the butter mixture, keeping the speed on low. Mix just until combined. Gather the dough into a ball and refrigerate to firm up just a bit, maybe 10-15 minutes. (Dough can be refrigerated overnight or for several days before baking it, too.)
Remove dough from the plastic wrap and roll out onto a floured cloth or board to a thickness of about 1/4-inch. Using a cookie cutter of any shape, or the top of a glass, cut the dough into the desired shapes. You may need to dip the cutters or glass top in flour between cutting every few cookies to keep the dough from sticking.
Place the dough onto the prepared baking sheet. Refrigerate on the baking sheet for 30 minutes. Remove from oven and bake for 20-25 minutes, or until tops are just beginning to turn golden-brown. Remove from oven and cool completely before icing and decorating.