Use a food processor or a mortar and pestle to grind the sugar and lavender petals together.
Put the sugar mixture and butter in the work bowl of a standing mixer that is outfitted with the paddle attachment. Beat at low speed until smooth -- about 2 minutes. Add the flour and beat until combined. Mixing is complete when there are no visible lumps of butter in the dough. Form the dough into a rough disk, wrap it in plastic, and chill for 30 minutes in the refrigerator.
Roll the dough into a 1/4-inch thick circle. Cut out cookie shapes with a round, 2-inch diameter cutter. Use a flat spatula to transfer the rounds to a parchment-lined baking sheet. Chill for 30 minutes. Meanwhile, preheat the oven to 300°F.
Bake on the middle rack of the preheated oven just until the sides of the cookies begin to color — 25-30 minutes. Let cool completely on the baking sheet.
Top the cookies with the glaze mixture. If desired, sprinkle fresh lavender petals over the cookies.