Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a large mixing bowl, cream together the softened butter and granulated sugar using an electric mixer or whisk until light and fluffy.
In a separate bowl, combine the all-purpose flour, cornstarch, dried lavender, lemon zest, and salt. Mix until well combined.
Gradually add the dry ingredients to the butter mixture, mixing until just combined; do not overmix.
Turn the dough out onto a floured surface and roll it out to about ½ inch thick. Use cookie cutters to cut out desired shapes or simply form small balls and flatten them slightly.
Place the cookies on the prepared baking sheet, leaving space between each cookie.
Bake in the preheated oven for 12-15 minutes, or until the edges are lightly golden. Remove from the oven and let them cool on the baking sheet for 5 minutes before transferring to a cooling rack.
To make the icing, whisk together the powdered sugar and lemon juice in a small saucepan until smooth. Drizzle over the cooled cookies.
If desired, garnish with additional lavender sprigs while the icing is still wet.