Measure 1 tablespoon dried culinary lavender and toast in a dry small skillet over medium-low heat for 30–45 seconds until fragrant; cool and chop finely. Preheat oven to 350°F and line a baking sheet with parchment.

In a large bowl, cream 1 cup room-temperature unsalted butter and 1 cup granulated sugar until light and fluffy, about 2–3 minutes; beat in 1 large egg, 1 teaspoon vanilla extract, 2 tablespoons fresh lemon juice, and zest of 2 lemons until combined.

Whisk together 2 1/4 cups all-purpose flour, 2 teaspoons cornstarch, 1/2 teaspoon baking soda, and 1/2 teaspoon salt in a separate bowl. Add the dry ingredients to the wet in two additions, mixing just until no streaks of flour remain. Fold in the toasted, chopped lavender.

Portion dough with a 1.5-tablespoon scoop onto the prepared sheet about 1 inch apart; gently flatten slightly if you prefer flatter cookies. Bake at 350°F for 10–12 minutes until edges are set and centers look slightly soft. Let rest on the sheet for 3 minutes, then transfer to a wire rack.

Whisk 1 cup powdered sugar with 2 tablespoons lemon juice, adding up to 1 more tablespoon if needed, to reach a smooth, spoonable glaze. While cookies are warm (about 5 minutes off the sheet), spoon or drizzle roughly 1/2 tablespoon glaze over each cookie and sprinkle reserved lavender if using. Let set 15–20 minutes before serving.