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, beat the butter and sugar until light and fluffy.
04 - Add eggs one at a time, then mix in lemon zest, lemon juice, and vanilla extract until combined.
05 - Alternate adding the flour mixture and milk to the wet ingredients, beginning and ending with the flour. Stir until just combined to avoid overmixing.
06 - Pour batter into prepared pan, smooth surface, and bake 28 to 32 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 wire rack to cool completely.
08 - Whisk powdered sugar with lemon juice until smooth and pourable. Drizzle evenly over the cooled cake.
09 - Cut into 8 equal slices and serve.