Preheat your oven to 350°F (177°C). Grease a 9x5-inch or 8x4-inch loaf pan (refer to the notes for guidance).
In a medium bowl, whisk together the all-purpose flour, baking powder, lemon zest, and salt. Set this mixture aside.
Using a stand mixer fitted with the paddle attachment (or an electric mixer), beat the butter and granulated sugar at medium-high speed for 4-6 minutes, until the mixture becomes pale and fluffy. Be sure to scrape down the sides of the bowl as needed.
Lower the speed to low and add the eggs one at a time, beating well after each addition. Incorporate the vanilla extract and lemon juice, then increase the speed to medium-high and beat until well combined.
With the mixer on low, add about one-third of the flour mixture, mixing until just nearly combined. Add half of the buttermilk, mixing gently until just combined. Repeat this process, adding another third of the flour mixture, followed by the remaining buttermilk, and ending with the final third of the flour. Mix only until the ingredients are just incorporated.
Transfer the batter into the prepared loaf pan and bake for 45-55 minutes, or until the cake turns golden brown and a toothpick inserted into the center comes out mostly clean, with just a few moist crumbs. Monitor the baking time closely.
Allow the cake to cool in the pan for around 15 minutes. Meanwhile, mix the lemon juice and powdered sugar to create the lemon syrup. Carefully remove the cake from the pan and place it on a cooling rack. While the cake is still warm, brush the lemon syrup over it, ensuring it soaks in well. Let the cake cool completely.
Once the cake has fully cooled, mix together the icing ingredients, starting with 1 tablespoon each of lemon juice and milk. Gradually add more lemon juice as needed to achieve a thick, but pourable consistency. Drizzle the icing over the cake and allow it to set before serving.