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 dry ingredients and the whole milk to the butter mixture, beginning and ending with the dry ingredients. 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 of the cakes comes out clean.
Remove the cakes from the oven and let them cool in the pans for 10 minutes.
Turn the cakes out onto wire racks and cool completely.
To prepare the lemon glaze, whisk together the powdered sugar and fresh lemon juice until smooth.
Once the cakes are completely cool, drizzle the lemon glaze over the top of the cakes or spread between the layers if stacking.