Chop lavender buds into smaller pieces. You can also use a food processor or mortar and pestle to break them up.
Press lavender and lemon zest into powdered sugar to release oils. Set aside.
In the bowl of a stand mixer or a large mixing bowl with a hand mixer, cream softened butter until smooth.
Add lavender-lemon-sugar mixture and cream for 2-3 minutes until smooth and well combined. Mix in vanilla extract.
Gradually add flour and mix until a dough forms. Scrape out onto a floured surface and use a floured rolling pin to roll out to ¼-inch thick.
Cut rounds with 2-inch round cookie cutters and place 2 inches apart on a lined cookie sheet. Place in freezer to chill 10-15 minutes while you preheat the oven to 325℉.
Bake cookies 12 minutes or until cookies are set. Allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.
In a medium bowl, whisk together lemon zest, lemon juice, and powdered sugar until smooth. If necessary, add a bit extra juice until a thick, but slight pourable consistency is reached.
Spread glaze over cooled cookies. If desired, top with a few lavender buds for decoration.
