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 until light and fluffy, about 2-3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the vanilla extract, fresh lemon juice, and lemon zest until combined.
Gradually add the dry ingredients to the wet ingredients, alternating with the whole milk, beginning and ending with the dry ingredients. Mix until just combined.
Pour the batter evenly into the prepared baking pan and smooth the top with a spatula.
Bake for 22-25 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool completely in the pan on a wire rack.
In a small bowl, whisk together the powdered sugar and 2 tablespoons fresh lemon juice until smooth to make the glaze.
Once the cake is completely cool, spread the lemon glaze evenly over the top.
Cut into 16 squares and serve.