In a medium mixing bowl, whisk together the all-purpose flour, cornstarch, lemon zest, and salt until the mixture is evenly blended.
In a large bowl, beat the softened butter with the powdered sugar on medium speed for 3 to 5 minutes, or until the mixture becomes pale and fluffy.
Add half of the dry mixture along with the lemon juice, and beat just until combined.
Incorporate the remaining dry ingredients, then switch to a wooden spoon or spatula and mix until a slightly sticky dough forms. Add 1 to 2 tablespoons of extra flour only if needed to achieve the proper consistency.
Transfer the dough to a large sheet of parchment paper and shape it into a log about 1 1/2 inches (4 cm) in diameter.
Wrap the dough in the parchment and refrigerate it for 1 hour, allowing it to firm up.
Remove the chilled dough from the refrigerator and slice it into rounds about 1/2 inch (1.3 cm) thick.
Arrange the slices carefully on a large parchment-lined baking sheet.
Place the sliced dough back into the refrigerator for 20 minutes while the oven preheats to 320°F (160°C).
Bake for 5 minutes, then raise the oven temperature to 350°F (180°C) and continue baking for 9 to 10 minutes more, until the cookies are lightly golden.
Leave the cookies on the baking sheet for about 5 minutes to set, then transfer them to a wire rack and cool completely.
For the glaze, whisk together the lemon juice, powdered sugar, and finely grated lemon zest in a small bowl until smooth.
Once the cookies have fully cooled, either dust them lightly with powdered sugar or spread them with the lemon glaze.