In the bowl of your stand mixer fitted with the paddle attachment, cream together the softened butter and granulated sugar on medium-high speed until light and fluffy, about 3-4 minutes. Scrape down the sides of the bowl as needed.

Add the eggs one at a time, beating well after each addition until fully incorporated. Stir in the lemon zest and vanilla extract until just combined.

In a separate medium bowl, whisk together the all-purpose flour, baking powder, and salt.

Gradually add the dry ingredients to the wet ingredients in three additions, mixing on low speed until just combined. Be careful not to overmix; stop when no dry streaks of flour remain.

Divide the dough in half, flatten each half into a disk, wrap tightly in plastic wrap, and refrigerate for at least 1 hour, or until firm enough to roll.

Preheat your oven to 375°F (190°C). Line baking sheets with parchment paper.

On a lightly floured surface, roll out one disk of dough to about ¼-inch thickness. Use your favorite cookie cutters to cut out shapes. Carefully transfer the cookies to the prepared baking sheets, leaving about 1-inch space between them.

Bake for 8-10 minutes, or until the edges are lightly golden brown. The centers should still appear slightly soft.

Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely. Repeat with the remaining dough.