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, beat the softened unsalted butter and granulated sugar together with an electric mixer on medium speed 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.
Gradually add the dry ingredients to the wet ingredients in three additions, alternating with the whole milk, beginning and ending with the dry ingredients. Mix until just combined. Do not overmix.
Pour the batter 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.
Run a knife around the edges to loosen the cake, then transfer it to a wire rack to cool completely before slicing into 8 equal pieces.