Preheat the oven to 350°F (175°C). Grease and flour an 8×8-inch 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 fresh lemon juice, lemon zest, and vanilla extract until combined.
Gradually add the dry ingredients to the wet ingredients alternately with the whole milk, beginning and ending with the dry ingredients. Mix until just combined.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 22 to 25 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and allow the cake to cool completely in the pan on a wire rack.
In a small bowl, whisk together the powdered sugar, fresh lemon juice, and lemon zest until smooth to make the glaze.
Once the cake has cooled, spread the lemon glaze evenly over the top.
Cut into 16 squares and serve.