Cream together room temperature unsalted butter and granulated sugar in a mixing bowl using an electric mixer until light and fluffy, about 3-5 minutes.

Add all-purpose flour, fresh lemon zest, and dried culinary lavender, mixing on low speed until just combined.

Fold in the egg until the dough is cohesive and smooth.

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

Preheat oven to 350°F (175°C). Roll out chilled dough on a floured surface to ¼ inch thick; cut into desired shapes.

Bake for 10-12 minutes until edges are slightly golden brown.

Cool cookies on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.

Combine powdered sugar, fresh lemon juice, and milk or cream in a bowl, whisk until smooth.

Dip cooled cookies into the icing or drizzle over them; sprinkle with dried lavender buds before icing sets.