In a large mixing bowl, using an electric mixer, beat 1 cup of room temperature unsalted butter and 3/4 cup of granulated sugar together on medium speed until light and fluffy, about 3 to 5 minutes.

Gradually add 2 cups of all-purpose flour, 1 tablespoon of fresh lemon zest, and 2 tablespoons of dried culinary lavender into the creamed mixture. Mix on low speed until incorporated.

Crack 1 large egg into the bowl and gently fold into the dough until cohesive.

Shape dough into a disc and wrap in plastic wrap. Refrigerate for at least 30 minutes.

Preheat your oven to 350°F (175°C).

Roll out the chilled dough to 1/4 inch thick and cut with cookie cutters. Place on parchment-lined baking sheets.

Bake for 10 to 12 minutes until edges are lightly golden.

Remove from oven and cool on baking sheet for 5 minutes before transferring to a wire rack.

Whisk together 1 cup of powdered sugar, 2 tablespoons of fresh lemon juice, and 1 to 2 tablespoons of milk until smooth for icing.

Dip tops of cooled cookies in icing or drizzle it over, and add lavender buds if desired.