Preheat the oven to 350°F (175°C). Grease and flour two 8-inch round cake pans.
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 3-4 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the lemon zest, fresh lemon juice, and vanilla extract until combined.
Alternately add the flour mixture and whole milk to the butter mixture, beginning and ending with the flour mixture. Mix each addition just until incorporated; do not overmix.
Divide the batter evenly between the prepared cake pans and smooth the tops with a spatula.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cakes from the oven and allow them to cool in the pans for 10 minutes.
Turn the cakes out onto wire racks to cool completely.
Prepare the lemon glaze by whisking together the powdered sugar, fresh lemon juice, and lemon zest in a small bowl until smooth.
Once the cakes are completely cooled, drizzle the lemon glaze over the top of the cakes, allowing it to drip down the sides.
Let the glaze set for about 15 minutes before slicing and serving.