Prep the Sugar: In a large bowl, combine the granulated sugar, lemon zest, and lavender. Use your fingers to rub the zest and lavender into the sugar until the sugar is fragrant and moist.
Cream: Add the butter to the sugar mixture and beat until pale and fluffy. Add the egg, vanilla, and lemon extract. Mix until combined.
Mix: Gradually stir in the flour until a soft dough forms.
Chill: Divide the dough, wrap in plastic, and chill in the fridge for 30–60 minutes.
Shape: Preheat your oven to 350°F (175°C). Roll the dough out on a floured surface to ¼-inch thickness and cut into your desired shapes.
Bake: Arrange on a lined baking sheet and bake for 10–12 minutes, or until the edges are just barely golden.
Glaze: Whisk the confectioners' sugar and lemon juice together. Once the cookies are completely cool, drizzle or dip them in the glaze. Let it set before serving.