In a small food processor, combine the granulated sugar, lemon zest, and basil leaves. Pulse and process until the sugar is very fragrant and resembles wet sand, and the basil pieces are very, very small. The sugar will be lime green in color. Resereve 2 tablespoons of the lemon basil sugar in a small bowl for dusting the cookies during baking – cover with plastic wrap to prevent the sugar from clumping.
In the bowl of a stand mixer, fitted with paddle attachment, cream the butter and salt over medium speed for until smooth and creamy. Add the remaining lemon basil sugar to the bowl with the butter, and beat over medium-low speed for 2 minutes, or until fluffy. Scrape down the bowl once or twice with a spatula to ensure that all of the ingredients are evenly incorporated.
On low speed, add the flour in two additions, mixing over low speed until the flour is just absorbed, scraping down the bowl once or twice. The dough will be shaggy and loose.
Scrape the dough onto a clean work surface. Using the “fraisage” technique, use the heel of your hand to press down and ‘smear’ the dough onto the countertop. Push back together and repeat until smooth and the dough comes together. Using a bench scraper (preferably one with a ruler for measuring), press the block of dough into a 5-inch square block. Wrap tightly in plastic wrap and refrigerate for 2 hours, or until firm.
Preheat the oven to 325°F (160°C), with one rack in the lower third and one rack in the upper third position in your oven. Line two sheet pans with parchment paper or silicone mats. Remove the dough from the refrigerator, remove the plastic wrap, and place the block between two sheets of parchment paper. Using a rolling pin, hit the dough evenly with a rolling pin, until it begins to flatten. Turn and repeat. By hitting the dough, you are warming and flattening it slowly, and preventing the dough from cracking. Roll the dough into a 9-inch square. If the dough gets too soft during this process, slide the parchment onto the back of a sheet pan and refrigerate until firm.
Using a large, sharp knife and ruler, score the dough along both edges of the dough to form 2¼-inch-long x 1½-inch-wide rectangles. Cut along the scores and transfer 12 rectangles to each sheet pan, setting the dough apart by at least one inch.
Sprinkle the tops of the dough lightly with the set aside lemon basil sugar. If the remaining sugar has started to clump, loosen it with your fingertips or re-process it in the food processor. Bake for 15 to 18 minutes, rotating and turning the pans halfway through, or until the edges of the shortbread are a pale golden brown. Allow the cookies to rest on the sheet pans for 5 minutes, than transfer to a wire rack to cool completely. Cookies can be stored in a covered container for up to 3 days.
Dough can be refrigerated for up to 2 days, or frozen for up to 1 month.