Preheat the oven to 350°F. Butter and line a 9x9 cake pan with parchment paper.
In a small bowl, combine one-third of the grated zest (from 1 lemon), ¼ cup sugar and a good pinch of flaky salt. Use your fingers and rub the mixture together until the sugar is tinted yellow and smells deeply of lemon. Set aside.
In a stand mixer fitted with the paddle attachment, beat the butter, powdered sugar, the remaining ¾ cup granulated sugar, the remaining lemon zest, kosher salt and vanilla on medium-high till it’s super light and fluffy, 3 to 5 minutes. In the beginning it will look like the cold butter won't break down, but give it time. Using a spatula, scrape down the sides of the bowl and, with the mixer on low, slowly add the flour and beat just to blend.
Pat the dough into the prepared pan, using your palm and dipping your fingers in a bowl of cold water. I also use a piece of parchment paper to smooth the surface. Sprinkle the top with the lemon-rubbed sugar, pressing it into the raw dough.
Bake until the top, bottom and edges are all nicely browned and the dough is set and firm to touch (but still slightly malleable), about 35 minutes. It is best to let the dough cool completely before removing.
When cooled, lift the shortbread out of the pan, cutting it into four equal squares, and then each square into four log bars. The bars will keep for days at room temperature, and will freeze for weeks after baking - wrap in plastic wrap and thaw to room temp before enjoying.