Step 1: In a large mixing bowl, cream together the softened unsalted butter and 1/2 cup of powdered sugar using an electric mixer until light and fluffy, about 2-3 minutes. Scrape down the sides of the bowl as needed.
Step 2: Beat in the lemon zest and vanilla extract until just combined.
Step 3: In a separate medium bowl, whisk together the all-purpose flour, cornstarch, and salt to ensure they are well combined.
Step 4: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix; stop as soon as no dry streaks of flour remain. The dough will be soft.
Step 5: Divide the dough in half, wrap each half tightly in plastic wrap, and chill in the refrigerator for at least 30 minutes, or up to 2 hours. This helps the dough firm up and prevents spreading during baking.
Step 6: Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper. Remove one half of the dough from the refrigerator. Roll the dough into small, 1-inch balls and place them about 1 inch apart on the prepared baking sheets.
Step 7: Bake for 10-12 minutes, or until the edges are just barely golden. The cookies should still appear pale in the center. Do not overbake. Let the cookies cool on the baking sheet for 5 minutes before carefully transferring them to a wire rack to cool completely.
Step 8: While the cookies are cooling, prepare the lemon glaze. In a small bowl, whisk together the remaining 1 cup of powdered sugar and 2-3 tablespoons of fresh lemon juice until a smooth, pourable glaze forms. Add more lemon juice if a thinner glaze is desired, or a tiny bit more powdered sugar if it's too thin.
Step 9: Once the cookies are completely cool, dip each cookie halfway into the lemon glaze, or drizzle the glaze over them. Place the glazed cookies back on the wire rack (with parchment paper underneath to catch drips) and let the glaze set completely, which takes about 30 minutes to an hour.