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