01 - Preheat oven to 350°F. Line an 8x8-inch baking pan with parchment paper or grease lightly.
02 - Whisk flour, baking powder, and salt together in a medium bowl; set aside.
03 - In a large bowl, cream softened butter and granulated sugar until light and fluffy.
04 - Beat in eggs one at a time, then add lemon zest, lemon juice, and vanilla extract; mix until combined.
05 - Alternately add flour mixture and milk to wet ingredients, starting and ending with flour; mix just until combined without overmixing.
06 - Pour batter into prepared pan and smooth surface. Bake for 28 to 32 minutes, or until a toothpick inserted in the center comes out clean.
07 - Allow cake to cool in pan for 10 minutes, then transfer to a wire rack and cool completely.
08 - Whisk powdered sugar and fresh lemon juice until smooth and pourable.
09 - Drizzle glaze over cooled cake, slice into 8 squares, and serve.