Step 1: In a large bowl, cream together the softened butter and granulated sugar with an electric mixer until light and fluffy (about 2-3 minutes).
Step 2: Beat in the eggs one at a time, mixing well after each addition. Stir in the vanilla extract and fresh lemon zest until just 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.
Step 5: Divide the dough in half, flatten each half into a disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes, or up to 2 hours.
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 disc of chilled dough to about 1/4-inch thickness. Use your favorite cookie cutters to cut out shapes. Reroll scraps as needed.
Step 8: Place cookies about 1 inch apart on the prepared baking sheets.
Step 9: Bake for 10-12 minutes, or until the edges are lightly golden and the centers are set. Let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Step 10: While the cookies cool, prepare the lemon glaze. In a medium bowl, whisk together the sifted powdered sugar and 3 tablespoons of fresh lemon juice until smooth. If the glaze is too thick, add more lemon juice (or milk) half a teaspoon at a time until you reach your desired consistency. If it's too thin, add more powdered sugar.
Step 11: Once cookies are completely cool, dip the tops into the glaze or drizzle it over them. Allow the glaze to set before storing or serving.