Preheat your oven to 350 and line two pans with parchment paper.
In a large mixing bowl, add your white sugar, brown sugar and butter and beat with a mixer till creamy. Add your lemon juice, lemon zest, egg and vanilla and mix together again briefly.
To the same bowl, sprinkle in your flour, baking soda, salt and cornstarch. Mix until the dough forms completely (it should be thick and smooth). If your dough isn't thick enough, add another 1-2 tbsp's of flour to your batter.
Using a small cookie scoop, scoop your dough onto your pans, spreading them apart evenly. About 12 cookies per each pan.
Bake in the oven for 10-12 minutes, or till the edges of the cookies are done. The tops may look slightly underdone but will continue to cook through as they cool.
Let your cookies cool on your pan for at least 15 minutes, and then remove them onto your cooling rack.
Store your cookies in an airtight container, and they taste best within the first 4 days.