In a large bowl, whisk together the flour, baking powder, and salt.
In a stand mixer fitted with the paddle attachment, cream together the butter and sugar on medium speed until light and fluffy, about 2 minutes. Scrape down the sides of the bowl with a rubber spatula.
Add the eggs, one at a time, mixing well between each addition and scraping the bowl. Mix in the vanilla, lemon extract, and zest (if using).
Add the flour mixture all at once and mix on low speed just until it all comes together. Scrape down the bowl, increase the speed to medium, and mix for 15 seconds.
Press the dough into a disk, wrap it in a plastic, and refrigerate until firm, at least 1 hour. It can be made a couple of days ahead or frozen for up to 3 weeks.
Preheat the oven to 350℉ (175℃). Line baking sheets with parchment paper.
Roll the dough out on a floured surface to a rectangle 1/8 inch / 3 mm thick.
Use cookie cutters to create fun shapes out of the dough. Place the cut cookies on the prepared baking sheets, leaving enough room for them to spread slightly, about 1 inch apart. If your dough starts to get warm and sticky, return it to the refrigerator, until firm, about 10 minutes. Warm dough will be difficult to work with.
Bake, one sheet at a time, in the middle of the oven for 8 to 12 minutes, until just golden on the bottom—you don't want too much color on them. The baking time will depend on the size of the cookie. Let them cool on the baking sheets completely before decorating.