Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking 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, cream the softened unsalted butter and granulated sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the freshly grated lemon zest, fresh lemon juice, and vanilla extract until combined.
Alternately add the dry flour mixture and whole milk to the wet ingredients, beginning and ending with the flour mixture. Mix until just combined.
Pour the batter evenly into the prepared baking pan and smooth the top with a spatula.
Bake in the preheated oven for 28 to 30 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing into 8 equal pieces.