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, whole milk, and vanilla extract until combined.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Pour the batter evenly into the prepared baking pan and smooth the top with a spatula.
Bake for 22 to 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, fresh lemon juice, and lemon zest until smooth to make the glaze.
Once the cake is cooled, spread the lemon glaze evenly over the top.
Allow the glaze to set for about 15 minutes before cutting the cake into 16 squares.
Serve and enjoy!