Cream softened butter, powdered sugar, and lemon extract together in the bowl of a stand mixer until smooth.
In a medium bowl whisk together flour, cornstarch, and salt.
Combine the dry ingredients with the wet ingredients until well combined by mixing in a stand mixer on low for 1 minute. (They should be mixed well but still crumbly)
Adjust the speed on the mixer to medium and mix for an additional 2-3 minutes. Until the dough comes together (mine came together around the 2 minute mark).
Roll the dough out into a 9"x12" rectangle on parchment paper (I like to use the parchment with grids). After it is 9"x12" transfer the parchment and dough to a baking sheet, cover with another sheet of parchment, and place in the refrigerator to cool for 30 minutes.
While the dough is cooling preheat your oven to 325º.
After refrigeration, lift the dough and parchment from the baking sheet and cut into 1 1/2" squares. (This should yield 48 cookies). Spread the cookies apart and place the parchment and cookies back onto the cookie sheet. These cookies maintain their shape very well so very little space is needed between the cookies. Just to be safe I still separated them onto 2 baking sheets.
Bake for 9-11 minutes or until their edges begin to turn a pale brown. Cool for one minute on the baking sheet and then transfer to a wire cooling rack to finish cooling.
After cookies are completely cooled combine powdered sugar, the juice from 1 lemon (about 3-4 Tbsp), and a tbsp of lemon zest together in a small bowl. This frosting will be drizzled onto the cookies. If necessary you can add more water (1 small drop at a time) to thin.
Using a spoon, drizzle the frosting over the cookies in a large sweeping motion.