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 infuse the sugar.
03 - In a medium bowl, whisk together flour, baking powder, baking soda, and salt.
04 - In a large bowl, cream the butter with the lavender-lemon sugar until light and fluffy. Beat in eggs, one at a time, followed by vanilla extract.
05 - Mix in the flour mixture in three additions, alternating with milk and lemon juice, starting and ending with flour. Do not overmix.
06 - Pour batter into the prepared pan and smooth the top. Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
07 - While the cake bakes, make the syrup: combine water, sugar, lemon juice, and lavender in a small saucepan. Bring to a simmer, stirring until sugar dissolves. Let steep 5 minutes, then strain.
08 - Cool cake in pan 10 minutes. Remove to a rack. While warm, brush with lemon-lavender syrup.
09 - When cake is completely cool, whisk glaze ingredients until smooth and drizzle over the cake.