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 with lavender and lemon oils.
03 - In a medium bowl, whisk together flour, baking powder, baking soda, and salt until well blended.
04 - In a large bowl, cream the butter with the lavender-lemon sugar until light and fluffy, approximately 3-4 minutes. 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. Mix until just combined, taking care not to overmix.
06 - Pour batter into the prepared pan and smooth the top with an offset spatula.
07 - Bake for 30-35 minutes, or until a wooden toothpick inserted into the center comes out clean.
08 - While the cake bakes, 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 through a fine-mesh sieve.
09 - Cool cake in pan for 10 minutes, then remove to a wire rack. While still warm, brush the lemon-lavender syrup evenly over the cake.
10 - When cake is completely cool, whisk powdered sugar, lemon juice, and milk until smooth. Drizzle over the cake and allow glaze to set before serving.