In the bowl of an electric mixer fitted with paddle attachment, beat the butter and sugar until light and fluffy, about 5 minutes.
Add in the lemon zest, rosemary, lemon extract, and vanilla extract. Beat for an additional minute.
Add in the eggs while the machine runs. The mixture will look slightly curdled.
Next, mix in the flour, baking powder, and salt. Stop as soon as the dry ingredients are incorporated and the dough comes together, ensuring not to overmix.
Turn out the dough and wrap it in plastic. Chill for at least 2 hours or overnight.
When ready to bake, preheat the oven to 350℉ and line two baking sheets with parchment paper.
Roll the dough out on a floured surface. It should be about 1/4-inch thick.
Use cookie cutters to cut the cookies into your desired shapes.
Transfer the cookies to the lined baking sheet, placing them about 2 inches apart. Bake for 8-10 minutes, removing them before the edges brown.
Allow to cool on the tray for 5 minutes before transferring to a wire rack to cool completely, about 1 hour.
Decorate as desired or enjoy plain.