In a mixing bowl, cream together the butter and granulated sugar until light and fluffy, about 3-5 minutes.

Gradually add the all-purpose flour, lemon zest, and dried lavender, mixing on low speed until incorporated. Fold in the egg using a spatula.

Shape the dough into a disc, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Preheat oven to 350°F (175°C). Roll out the dough to about ¼ inch thickness and cut out shapes.

Place cookies on a baking sheet lined with parchment paper and bake for 10-12 minutes until edges are lightly golden.

Remove from oven, let cookies rest on baking sheet for 5 minutes, then transfer to a wire rack to cool completely.

In a bowl, whisk together the powdered sugar, lemon juice, and milk until smooth. Adjust consistency as needed.

Dip cooled cookies in icing or drizzle over them, then sprinkle with lavender before icing sets.