Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper.

In a mixing bowl, beat together the softened unsalted butter and granulated sugar for about 3-5 minutes until fluffy.

Gradually sift in the all-purpose flour, lemon zest, and dried lavender, mixing on low speed until just combined.

Add the large egg and mix on low speed until no dry flour remains.

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

Roll out the chilled dough to about 1/4 inch thickness and cut out cookies with cookie cutters.

Bake for 10-12 minutes until edges are golden and centers are light. Cool on baking sheets before transferring to a wire rack.

Prepare icing by whisking together powdered sugar, lemon juice, and milk until smooth.

Once cookies are cooled, drizzle or dip the tops into the icing and let set.