Step 1: In a large bowl, cream together the softened butter and granulated sugar until light and fluffy, about 2-3 minutes.
Step 2: Beat in the egg, vanilla extract, and fresh lemon zest until well combined.
Step 3: In a separate medium bowl, whisk together the all-purpose flour, baking powder, and salt.
Step 4: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix. The dough will be soft.
Step 5: Divide the dough in half, flatten each half into a disk, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes (or up to 2 hours) to firm up.
Step 6: Preheat your oven to 375°F (190°C). Line baking sheets with parchment paper.
Step 7: On a lightly floured surface, roll out one disk of chilled dough to about 1/4-inch thickness. Cut out desired shapes with cookie cutters.
Step 8: Place cookies about 1 inch apart on the prepared baking sheets.
Step 9: Bake for 8-10 minutes, or until the edges are lightly golden and the centers are just set. Do not overbake.
Step 10: Let cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.
Step 11: While cookies cool, prepare the glaze: In a medium bowl, whisk together the sifted powdered sugar and 3 tablespoons of fresh lemon juice until smooth. Add more lemon juice a teaspoon at a time if a thinner consistency is desired.
Step 12: Once cookies are completely cool, dip the tops into the lemon glaze or spread the glaze with a small offset spatula. Let the glazed cookies set on a wire rack over parchment paper before serving.