In a mason jar or bowl with a tight fitting lid, mix together the granulated sugar and lemon zest. Cover tightly and set aside.
½ cup granulated sugar, 1 teaspoon lemon zest
In a stand mixer fitted with a paddle attachment, cream the butter, granulated sugar, powdered sugar, and salt until thoroughly combined, about 3 minutes.
1 cup unsalted butter, ¼ cup granulated sugar, ¼ cup powdered sugar, ¼ teaspoon kosher salt
In a separate bowl, sift together the all-purpose and cake flours.
1 cup all-purpose flour, 1 cup cake flour
With the mixer on low, mix the flour mixture into the butter mixture in 3 additions, scraping down the sides of the bowl in between each addition.
Add the lavender and lemon zest and stir just until combined.
1 tablespoon fresh lavender , 1 teaspoon lemon zest
Roll out the dough to a ½-inch-thick rectangle. Wrap tightly in plastic wrap and refrigerate for at least 4 hours or overnight.
Preheat the oven to 350ºF. Cut the refrigerated dough into 2 x 1-inch rectangular bars and place on a parchment-lined baking sheet. Bake the cookies until just golden at the edges, about 20 minutes. Remove from the oven and immediately sprinkle with lemon sugar. Let cool completely before serving.