Zest the lemons before juicing them, then measure out 2 tbsp zest and 2 tbsp juice.
In a mixing bowl, whisk together the flour, cornstarch, and salt.
In a separate bowl, whisk the olive oil, 80 g sugar, lemon zest, lemon juice, and vanilla for about 1 to 2 minutes, until the mixture looks slightly thick and well combined.
Add the dry mix to the wet mix and fold with a spatula until the dough comes together with no dry streaks. The dough will feel soft and a little oily, not sticky like a cookie dough.
Turn the dough onto a sheet of plastic wrap and shape it into a log about 5 cm / 2 inches wide. Wrap tightly and chill for 45 minutes to 1 hour, until firm enough to slice cleanly.
Unwrap the log and roll it in the remaining 2 tbsp sugar, pressing gently so the sugar sticks to the outside.
Heat the oven to 170 C / 340 F and line a sheet pan (33 x 46 cm / 13 x 18 inch) with parchment paper.
Slice the log into rounds about 1 cm / 0.4 inch thick with a sharp knife. If the dough crumbles instead of slicing clean, let it sit at room temperature for 2 to 3 minutes and try again.
Arrange the rounds on the pan, spacing them about 2.5 cm / 1 inch apart.
Bake for 14 to 16 minutes, until the edges turn light gold and the centers still look pale and set.
Let the cookies cool on the pan for 5 minutes, then move them to a wire rack to cool completely. They firm up as they sit.