01 - Preheat oven to 350°F. Grease and line an 8-inch square cake pan with parchment paper.
02 - Whisk together flour, baking powder, and salt in a medium bowl; set aside.
03 - Beat softened butter and granulated sugar in a large bowl until light and fluffy, about 2 to 3 minutes.
04 - Incorporate eggs one at a time, mixing well after each. Add lemon zest, lemon juice, and vanilla extract; blend thoroughly.
05 - Alternate adding dry ingredients and milk to the butter mixture, starting and ending with the dry. Mix gently until just combined without overmixing.
06 - Pour batter into prepared pan, smoothing the surface, and bake for 30 to 35 minutes until a toothpick inserted in the center comes out clean.
07 - Allow cake to cool in the pan for 10 minutes, then transfer to a cooling rack until completely cooled.
08 - Beat softened butter until creamy. Gradually add powdered sugar, then mix in lemon juice, lemon zest, vanilla extract, and milk to achieve a spreadable consistency.
09 - Evenly spread frosting over cooled cake. Slice into 8 pieces and serve.