01 - Preheat the oven to 350°F. Line an 8x8-inch baking pan with parchment paper or lightly grease it.
02 - Whisk together flour, baking powder, and salt in a medium bowl. Set aside.
03 - In a large bowl, cream the softened butter and granulated sugar until light and fluffy.
04 - Beat in eggs one at a time, then add lemon zest, fresh lemon juice, and vanilla extract. Mix until evenly combined.
05 - Alternately add the flour mixture and milk to the wet ingredients, starting and ending with the flour. Mix gently until just combined to avoid overmixing.
06 - Pour the batter into the prepared pan, smoothing the surface. Bake for 28 to 32 minutes, or until a toothpick inserted in the center comes out clean.
07 - Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack and cool completely.
08 - Whisk together powdered sugar and fresh lemon juice until smooth and pourable.
09 - Drizzle the lemon glaze over the cooled cake, slice into 8 squares, and serve.