Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a mixing bowl, cream together the softened butter and granulated sugar until light and fluffy using an electric mixer or whisk.
Add the lemon zest, vanilla extract, and dried lavender to the butter mixture and mix until well combined.
In a separate bowl, whisk together the all-purpose flour, cornstarch, and salt. Gradually add the dry ingredients to the wet mixture, mixing until a dough forms.
Roll out the dough on a lightly floured surface to about ¼ inch thickness. Use cookie cutters to cut out shapes, or simply cut into squares if preferred.
Place the cookies on the prepared baking sheet and bake for 12-15 minutes, or until the edges are lightly golden.
Remove from the oven and let the cookies cool on a cooling rack.
While the cookies cool, prepare the icing by mixing the powdered sugar and lemon juice until smooth. Adjust the consistency as necessary by adding either more powdered sugar or lemon juice.
Drizzle the icing over the cooled cookies and garnish with additional lavender if desired.
Allow the icing to set before serving.