Preheat the oven to 325 F. Line a 9×5 inch loaf pan with parchment paper.
In a large bowl, combine sugar and lemon zest. Mix together thoroughly to extract the oil from the zest into the sugar. Add the softened butter and mix until fully incorporated. Add the vegetable oil and whisk together until the mixture is airy and lighter in color. Add the eggs, vanilla, lemon juice, and buttermilk then whisk together.**
In a separate bowl, whisk together flour, salt, baking powder, and baking soda. Combine the dry ingredients into the wet ingredients. Whisk together just until it becomes smooth. Do not overmix.
Pour batter into a lined loaf pan and bake in the oven for 40-50 minutes. Remove from oven and let cool on a wire rack for at least an hour.
When the cake is cooled completely, mix together milk, lemon juice, and powdered sugar to make the icing. Pour icing over the cake while it’s still in the pan and let sit for 4-5 minutes to set. Remove from the pan and enjoy.