1. Preheat oven to 350°F (177°C). Grease a 9×5-inch or 8×4-inch loaf pan thoroughly or line with parchment paper to prevent sticking.
2. In a medium bowl, whisk together the flour, baking powder, lemon zest, and salt; set aside.
3. Using a stand mixer or electric mixer, cream the butter (and vegetable oil if using) with sugar on medium-high speed until pale and fluffy, about 4-6 minutes, scraping the bowl sides as needed.
4. On low speed, add eggs one at a time, beating well after each addition. Then mix in vanilla extract and lemon juice.
5. Alternately add the dry flour mixture and the buttermilk in thirds: start with one-third of the flour mixture, mix until just combined, add half the buttermilk, mix briefly; repeat with the next third of flour, remaining buttermilk, and last third of flour, mixing only until incorporated to avoid overmixing.
6. Pour batter into prepared pan and bake for 45-55 minutes, or until golden brown and a toothpick inserted into the center comes out mostly clean with a few moist crumbs.
7. Let the cake cool in the pan for 15 minutes. Mix lemon juice and powdered sugar for the syrup and brush it over the warm cake to soak in.
8. Remove cake to a wire rack and cool completely.
9. Prepare icing by combining powdered sugar, lemon juice, and milk, starting with 1 tablespoon lemon juice and milk and adding remaining lemon juice as needed to achieve a thick but pourable glaze. Pour over the cooled cake and let dry before serving.