Set the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a large bowl, beat softened butter with granulated and powdered sugar until pale and fluffy.
Mix in lemon zest and finely chopped lavender until evenly combined.
Whisk flour and salt in a separate bowl. Gradually add to the butter mixture, mixing until a soft dough forms. Use your hands to bring it together if needed.
Roll into a 2-inch thick log, wrap in plastic, and chill for 30 minutes. Or roll the dough flat to ¼ inch and use cookie cutters.
Slice log into ¼-inch rounds or cut out shapes. Place on baking sheet, spaced apart.
Bake for 10–12 minutes, until the edges are lightly golden. Cool on the sheet for a few minutes, then transfer to a wire rack.
Whisk powdered sugar, lemon juice, and vanilla until smooth. Adjust with more lemon juice or sugar for desired consistency.
Drizzle or spread icing on cooled cookies. While wet, sprinkle with crushed lavender if desired.