In a food processor , process the sugar, lemon zest and lavender until the sugar looks damp and the zest and lavender are fully incorporated, about 30 seconds. Whisk together the flour, baking soda and salt, then add to the sugar mixture; pulse to combine, about 10 1-second pulses.
Scatter the butter pieces over and pulse until the mixture resembles fine cornmeal, about 15 1-second pulses. In a measuring cup, beat together the lemon juice, egg yolk and vanilla. With the machine running, add the juice mixture in a slow stream (should take about 10 seconds) and continue processing until the dough forms into a ball, 10 to 15 seconds longer.
Turn the dough and any dry bits onto a clean work surface and gently gather into a ball. Working quickly, roll the dough into a cylinder about 12 inches; long and 1½” in diameter. Center the dough on a piece of parchment or plastic wrap and wrap tightly, twisting the ends together to seal. Chill the dough until firm and cold, about 45 minutes in the freezer or 2 hours in the refrigerator.
Preheat the oven to 375°F Line baking sheets with parchment paper or spray them with nonstick cooking spray. Seriously. Even if you think your pans are non-stick pans, and even if you’re out of parchment paper and don’t feel like running to the store in the middle of all this. Line the sheets with parchment paper! Otherwise your delicate cookies will stick to the pan and crumble when you try to remove them.
Remove the dough log from its wrapper and use a sharp knife to slice the dough into 3/8 inch; thick rounds. Place the rounds on the prepared baking sheets, spacing them 1 inch; apart. Bake until the centers of the cookies just begin to color and the edges are golden brown, about 10-12 minutes (the original recipe says 14-16 minutes, but 10 minutes was perfect for me, rotating the baking sheet halfway through the baking time. Cool on sheets about 5 minutes, then transfer the cookies to a wire rack to cool completely.