Preheat your oven to 350°F (175°C). Lightly grease and flour an 8x8 or 9x9 inch square baking pan, or line it with parchment paper, leaving some overhang on the sides for easy lifting. I always prefer lining with parchment; it makes getting the cake out a breeze!
In your nice big mixing bowl, whisk together the all-purpose flour, granulated sugar, powdered sugar, baking powder, and salt. Give it a good mix until everything is well combined and there are no clumps of powdered sugar.
Now, pour in the melted butter. Add the eggs, whole milk, lemon zest, and ALL of the freshly squeezed lemon juice (that's the 1/4 cup PLUS the 2 tablespoons).
Using a whisk or a rubber spatula, stir everything together until just combined. Don't overmix! A few small lumps are totally okay. Overmixing can lead to a tough cake.
Pour the batter evenly into your prepared pan. Give the pan a gentle tap on the counter to help settle the batter and release any air bubbles.
Bake for 30-40 minutes, or until a wooden skewer or toothpick inserted into the center comes out clean. The top should look golden brown and feel set.
Let the cake cool in the pan on a wire rack for at least 15-20 minutes before slicing and serving. This helps it set up properly.
Once slightly cooled, you can lift it out using the parchment paper handles (if you used them) and slice it into squares or bars.