In a large bowl, rub the lemon zest into the granulated sugar with your fingertips for about 1 minute, until fragrant and slightly damp. This releases the lemon oils for a much brighter flavor.
Add the butter and beat with a mixer on medium speed for 2-3 minutes, until light and fluffy. Beat in the Greek yogurt, lemon juice and vanilla until smooth.
In a separate bowl, whisk together the flour, cornstarch, baking powder, baking soda and salt. Add to the butter mixture and mix on low just until a soft dough forms.
Cover and chill the dough for 30 minutes. Near the end, preheat the oven to 350°F and line two baking sheets with parchment paper.
Roll the dough into 1 1/2 tbsp balls, place them 2 inches apart and flatten each one slightly with your palm.
Bake for 10-11 minutes, until the bottoms are lightly golden and the tops are set but still pale.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely. The glaze sets best on fully cooled cookies.
Whisk the powdered sugar, lemon juice and lemon zest until smooth. Dip the top of each cookie into the glaze, let the excess drip off and let set for 20 minutes. Makes about 20 cookies.