Preheat the Oven: Set the oven to 350°F (177°C). Grease a 9x5-inch (or 8x4-inch) loaf pan. (Note: If you're unsure about your pan's performance, line it with parchment paper.)
Mix the Dry Ingredients: In a medium bowl, combine the flour, baking powder, lemon zest, and salt. Set aside.
Cream the Butter and Sugar: In a stand mixer fitted with the paddle attachment (or using an electric mixer), cream the butter and sugar together on medium-high speed until pale and fluffy, about 4-6 minutes. Scrape the sides of the bowl as needed.
Add Eggs and Flavors: With the mixer running on low-speed, add the eggs one at a time. Then beat in the vanilla extract and lemon juice. Increase the speed to medium-high and beat until well combined.
Combine Dry and Wet Ingredients: With the mixer on low-speed, add about one-third of the flour mixture, mixing until almost incorporated. Add half of the buttermilk and mix until just combined. Repeat with another third of the flour mixture, then the remaining buttermilk, and finish with the last third of the flour mixture. Beat until just combined.
Bake the Cake: Scrape the batter into the prepared loaf pan. Bake for 45-55 minutes, until the cake is golden brown, and a toothpick inserted comes out mostly clean with just a few moist crumbs. Baking times may vary, so keep an eye on your cake.
Make the Lemon Syrup: While the cake is baking, mix the lemon juice and powdered sugar for the lemon syrup.
Cool and Brush with Syrup: Let the cake cool in the pan for about 15 minutes. Carefully invert the pan to transfer the cake to a cooling rack. While the cake is still warm, brush the lemon syrup over it. Allow the cake to cool completely.
Prepare the Lemon Icing: Once the cake is fully cooled, combine the powdered sugar, lemon juice, and milk. Start with 1 tbsp of lemon juice and milk, then add the remaining lemon juice as needed. The icing should be thick and not runny.
Ice the Cake: Pour the icing over the cooled cake and let it set before serving.