Position your oven racks to divide the oven into thirds and preheat it to 350 degrees F.
Line two baking sheets with parchment paper.
Whisk the flour, baking soda and baking powder together.
Grate the zest of 1 lemon and squeeze the juice. If you dont have 1/4 cup after the first lemon, squeeze the juice from the second lemon.
Put the sugar and lemon zest in the bowl of a stand mixer.
Using your fingertips, rub tthe ingredients together until the sugar is moist and fragrant.
Add the butter and salt to the bowl and beat on medium speed until the mixture is smooth, about 2 minutes. Beat in the egg, followed by the vanilla and lemon juice.
Turn off the mixer, add half of the dry ingredients and mix on low speed until they're almost incorporated. Scrape down the sides and bottom of the bowl with a spatula.
Add the rest of the flour and beat on low speed until the dough comes away from the sides of the bowl.
Roll each portion into a ball between your palms, drop into the sugar, roll it around to coat and place on the baking sheets. These cookies spread dramatically, so make sure to leave about 2 inches between them.
Bake the cookies for 8 to 14 minutes. If you bake them for 8 to 10 minutes, you'll get pale cookies that will be chewy; bake them for 12 to 14 minutes, until theyre barely golden around the edges (the bottoms will be lightly browned), and you'll get cookies that are chewy in the center and crisp around the edges. I baked mine for 12 minutes and enjoyed the crispy/chewy version.
The cookies will be crackle-topped and too soft to lift from the baking sheets after cooking. Transfer the whole baking sheet to cooling racks and let the cookies cool completely before you move them.
Adapted from Dorie's Cookies