01 - Preheat oven to 350°F. Grease and flour a 9-inch round cake pan or line with parchment paper.
02 - In a small bowl, combine sugar, lavender buds, and lemon zest. Rub together with fingers to release oils and infuse the sugar.
03 - In a medium bowl, whisk together flour, baking powder, baking soda, and salt until well blended.
04 - In a large bowl, beat softened butter with the lavender-lemon sugar until light and fluffy, about 3 minutes. Add eggs one at a time, beating well after each addition, then mix in vanilla extract.
05 - Add flour mixture in three additions, alternating with milk and lemon juice. Begin and end with flour addition. Mix until just combined, taking care not to overmix.
06 - Pour batter into the prepared pan and smooth the top. Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
07 - While the cake bakes, combine water, sugar, lemon juice, and lavender in a small saucepan. Bring to a simmer over medium heat, stirring until sugar completely dissolves. Remove from heat and let steep for 5 minutes, then strain through a fine-mesh sieve.
08 - Allow the cake to cool in the pan for 10 minutes. Transfer to a wire rack. While still warm, brush the lemon-lavender syrup evenly over the top and sides.
09 - Once the cake has cooled completely, whisk powdered sugar, lemon juice, and milk until smooth and pourable. Adjust consistency with additional milk if needed. Drizzle glaze over the cake allowing it to cascade down the sides.