01 - Preheat oven to 350°F. Grease and line an 8-inch square cake pan with parchment paper.
02 - Whisk flour, baking powder, and salt together in a medium bowl. Set aside.
03 - Beat softened butter and granulated sugar until light and fluffy, about 2 to 3 minutes.
04 - Add eggs one at a time, beating well after each. Mix in lemon zest, lemon juice, and vanilla extract.
05 - Alternately add dry ingredients and milk to the butter mixture, beginning and ending with dry ingredients. Mix until just combined, avoiding overmixing.
06 - Pour batter into prepared pan and smooth the surface evenly.
07 - Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
08 - Remove from oven; cool in pan for 10 minutes. Transfer cake to a wire rack and cool completely.
09 - Beat softened butter until creamy. Gradually add powdered sugar and mix well. Incorporate lemon juice, zest, vanilla, and milk until smooth and spreadable.
10 - Spread frosting evenly over cooled cake. Slice into 8 portions and serve.