Preheat the oven to 350°F (175°C). Grease and flour a 9×5-inch loaf pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, beat the softened unsalted butter and granulated sugar on medium speed until light and fluffy, about 4-5 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the freshly grated lemon zest and pure vanilla extract until combined.
Reduce the mixer speed to low and alternately add the dry flour mixture and whole milk to the butter mixture, beginning and ending with the flour mixture. Mix until just combined.
Stir in the fresh lemon juice by hand with a spatula.
Pour the batter evenly into the prepared loaf pan and smooth the top with a spatula.
Bake in the preheated oven for 55 to 65 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 15 minutes.
Run a knife around the edges to loosen the cake, then invert it onto a wire rack to cool completely before slicing and serving.