Preheat your oven to 350°F (177°C). Lightly grease a 9x5-inch (or 8x4-inch) loaf pan, or line it with parchment paper. Set aside.

In a medium bowl, whisk together the flour, baking powder, lemon zest, and salt. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment (or using a handheld electric mixer), cream the butter and sugar on medium-high speed until the mixture is light, pale, and fluffy—this should take about 4 to 6 minutes. Scrape down the sides of the bowl as needed to ensure even mixing.

With the mixer on low speed, add the eggs one at a time, mixing well after each addition. Then beat in the vanilla extract and lemon juice. Increase to medium-high speed and beat until fully combined and smooth.

Reduce the mixer speed to low. Add one-third of the flour mixture, mixing just until nearly combined. Then add half of the buttermilk, again mixing just until incorporated. Repeat with another third of the flour, followed by the remaining buttermilk, and finish with the last third of the flour. Be careful not to overmix—blend only until everything is just combined.

Pour the batter into the prepared loaf pan, smoothing the top with a spatula. Bake for 45 to 55 minutes, or until the cake is golden brown and a toothpick inserted into the center comes out with just a few moist crumbs. Baking time may vary depending on your oven and pan size, so begin checking around the 45-minute mark.

Let the cake cool in the pan for 15 minutes. While it cools, stir together the lemon juice and powdered sugar to make the lemon syrup.

Carefully remove the cake from the pan and transfer it to a wire rack. While the cake is still warm, brush the lemon syrup over the top and sides, allowing it to soak in as the cake cools completely.

Once the cake is fully cooled, prepare the lemon icing by mixing the powdered sugar, 1 tablespoon lemon juice, and milk. Add additional lemon juice as needed until the icing is thick but pourable—not runny. Drizzle the icing over the cake, letting it set before slicing and serving.