01 - In a large mixing bowl, beat the softened butter and powdered sugar together using an electric mixer on medium speed until the mixture becomes pale, light, and fluffy, approximately 3 to 4 minutes.
02 - Add the lemon zest, chopped dried lavender, and salt to the butter mixture. Mix on low speed until evenly distributed throughout the dough, scraping down the sides of the bowl as needed.
03 - Gradually add the all-purpose flour to the bowl, mixing on low speed just until the dough comes together and no visible dry streaks remain. Be careful not to overmix.
04 - Divide the dough into two equal portions. Shape each portion into a uniform log approximately 2 inches in diameter. Wrap each log tightly in plastic wrap and refrigerate for at least 30 minutes or until firm enough to slice cleanly.
05 - Preheat the oven to 350°F. Line two large baking sheets with parchment paper and set aside.
06 - Remove the chilled dough logs from the refrigerator. Using a sharp knife, slice each log into rounds approximately 1/4-inch thick. Arrange the cookie slices on the prepared baking sheets, leaving about 1 inch of space between each cookie.
07 - Bake the cookies for 14 to 16 minutes, rotating the baking sheets halfway through, until the edges are lightly golden. The centers should remain pale. Let the cookies cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.
08 - While the cookies cool, whisk together the powdered sugar, 2 tablespoons of lemon juice, and lemon zest in a small bowl until smooth and glossy. If the icing is too thick to drizzle, add additional lemon juice 1 teaspoon at a time until the desired consistency is reached.
09 - Once the cookies have cooled completely, drizzle or spread the lemon icing over each cookie. If desired, sprinkle a few lavender buds on top of the icing while it is still wet. Allow the icing to set for at least 15 minutes before serving or storing.