Preheat oven to 350 degrees F. Grease and flour 2 (8 ½ by 4 ¼ by 2 ½- inch) loaf pans.
Cream the butter and 2 cups cane sugar in the bowl of an electric mixer fitted with the paddle attachment, until light and fluffy, about 5 minutes. With the mixer on medium speed, add the eggs, 1 at a time, then add lemon zest
Sift together the flour, baking powder, baking soda, and salt in a bowl. In another bowl, combine ¼ cup lemon juice, the buttermilk, and vanilla. Add the flour and buttermilk mixtures alternately to the batter, beginning and ending with the flour. Gently stir in the lavender buds. Divide the batter evenly between the pans, smooth the tops, and bake for 45 minutes to 1 hour, until a cake tester comes out clean.
Combine ½ cup granulated sugar with ½ cup lemon juice in a small saucepan and cook over low heat until the sugar dissolves. When the cakes are done, allow to cool for 10 minutes. Remove the cakes from the pans and set them on a rack set over a parchment lined sheet pan. Holding the warm cakes upside down, use a pick to pierce the bottom all around, carefully spoon the lemon syrup over the bottom of the cakes allowing it to seep into the holes you have just created. Allow to cool completely.
Combine the confectioners’ sugar and the lemon juice in a bowl, using a whisk, mix until smooth. Pour over the tops of the cakes, allowing the glaze to drizzle down the sides.