01 - Place softened unsalted butter and powdered sugar in a large mixing bowl. Beat with an electric mixer on medium-high speed for 3 to 4 minutes until the mixture becomes pale, fluffy, and significantly increased in volume.
02 - Add the freshly grated lemon zest, finely chopped dried lavender, and sea salt to the creamed butter mixture. Continue mixing on low speed until evenly distributed throughout the dough, about 30 seconds.
03 - With the mixer running on low speed, gradually add the all-purpose flour in three additions. Mix just until a cohesive dough forms and no dry streaks remain. Avoid overmixing to prevent tough cookies.
04 - Divide the dough into two equal portions. On a clean surface, roll each portion into a uniform log approximately 2 inches in diameter. Wrap each log tightly in plastic wrap and refrigerate for a minimum of 30 minutes or up to 24 hours until completely firm.
05 - Position oven racks in the upper and lower thirds. Preheat to 350°F. Line two large rimmed baking sheets with parchment paper.
06 - Remove one chilled dough log at a time from the refrigerator. Using a sharp knife, slice crosswise into rounds about 1/4 inch thick. Arrange cookies on prepared baking sheets, spacing them 1 inch apart to allow for minimal spreading.
07 - Bake for 14 to 16 minutes, rotating the pans halfway through, until edges are barely golden and centers appear set. The cookies should remain pale. Let cool on the baking sheets for 5 minutes before transferring to a wire rack.
08 - While cookies cool, whisk together powdered sugar, 2 tablespoons lemon juice, and lemon zest in a small bowl until smooth and glossy. If the glaze is too thick to drizzle, add additional lemon juice 1 teaspoon at a time until desired consistency is reached.
09 - Once cookies have completely cooled, drizzle or spread the lemon glaze over each cookie using a small spoon or offset spatula. Immediately sprinkle a few whole lavender buds on top for decoration. Allow glaze to set for at least 15 minutes before serving.