Step 1Heat the oven to 350 degrees. Butter and flour a 9x5-inch loaf pan.
Step 2Melt the butter with 1 tablespoon of the lavender in a small saucepan. Let the mixture steep 10 minutes, then strain, discarding the lavender. Set aside to cool.
Step 3Beat the eggs and 1 cup of the sugar in the bowl of a mixer until thick and pale, about 5 minutes.
Step 4Sift together the flour and salt into a bowl. Using a whisk, fold the lemon zest and one-third of the flour mixture into the eggs until thoroughly combined. Fold in the rest of the flour in 2 batches. In a separate bowl, whisk 1 cup of the batter with the melted butter and the vanilla. Add this to the remaining batter and fold to combine. Pour the batter into the prepared pan.
Step 5Bake the cake until a toothpick inserted in the middle comes out clean, 40 to 45 minutes. If the top of the cake seems to be getting overly browned before the center is set, cover with foil and continue baking.
Step 6Meanwhile, combine the remaining 1/2 cup of sugar, the lemon juice, 1/4 cup of water and the remaining 3 tablespoons of lavender in a saucepan. Bring the mixture to a simmer and cook, stirring, until the sugar dissolves.
Step 7Transfer the cake to a wire rack. Using a cake tester or skewer, poke the cake all over. Brush the loaf with half of the syrup and let cool for 10 minutes. Invert the cake onto the rack, remove the pan and brush the syrup over the bottom and sides of the cake. Turn the cake back over and brush with the remaining syrup. Let cool completely.