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 - Using a mixer, beat butter and sugar in a large bowl 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 - Alternate adding flour mixture and milk to the wet ingredients, beginning and ending with the flour. Mix until just combined without overmixing.
06 - Pour batter into prepared pan and smooth the surface. Bake 28 to 32 minutes until a toothpick inserted in center comes out clean.
07 - Allow cake to cool in pan for 10 minutes, then transfer to wire rack to cool completely.
08 - Whisk powdered sugar and lemon juice until smooth and pourable. Drizzle glaze over cooled cake.
09 - Slice into 8 squares and serve.