In a large mixing bowl, cream together the softened butter and powdered sugar until light and fluffy.
Add the vanilla extract, lemon zest, and lemon juice, and mix until smooth.
In a separate bowl, whisk together the flour, salt, and finely chopped lavender.
Gradually add the dry ingredients to the butter mixture, mixing until a soft dough forms.
Shape the dough into a log, wrap it in plastic wrap, and refrigerate for 1 hour. Alternatively, you can roll out the dough and cut it into shapes before chilling.
Preheat the oven to 350°F (175°C). Slice the chilled dough into 1/4-inch thick rounds or cut shapes.
Place on a parchment-lined baking sheet and bake for 10–12 minutes, or until the edges are lightly golden.
Allow cookies to cool completely before icing.
In a small bowl, whisk together the powdered sugar, lemon juice, lemon zest, and milk until smooth, adjusting the consistency as needed.
Drizzle or spread the icing over the cooled cookies and sprinkle with dried lavender for garnish.
Let the icing set for 15–20 minutes before serving.