01 - Preheat oven to 350°F. Grease and flour a 9-inch round cake pan or line with parchment paper.
02 - Combine sugar, lavender buds, and lemon zest in a small bowl. Rub together with fingers to release essential oils and infuse the sugar.
03 - Whisk together flour, baking powder, baking soda, and salt in a medium bowl until thoroughly combined.
04 - Beat softened butter with lavender-lemon sugar in a large bowl until light and fluffy. Add eggs one at a time, mixing well after each addition. Stir in vanilla extract.
05 - Add flour mixture in three additions, alternating with milk and lemon juice. Begin and end with flour. Mix until just combined, being careful not to overmix.
06 - Pour batter into prepared pan and smooth surface. Bake for 30-35 minutes until a toothpick inserted into center comes out clean.
07 - While cake bakes, combine water, sugar, lemon juice, and lavender in a small saucepan. Bring to simmer, stirring until sugar dissolves. Remove from heat and let steep for 5 minutes, then strain.
08 - Cool cake in pan for 10 minutes, then transfer to wire rack. Brush warm cake generously with lemon-lavender syrup.
09 - When cake is completely cool, whisk powdered sugar, lemon juice, and milk until smooth. Drizzle glaze over cake and allow to set before serving.