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 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 flour mixture to the wet ingredients, alternating with the whole milk, 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 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 is completely cool, spread the lemon glaze evenly over the top.
Cut into 16 squares and serve.