1. Preheat the oven to 350°F (175°C). Grease a 9×13-inch baking pan and line it with parchment paper.

2. Whisk together the flour, baking powder, and salt in a medium bowl.

3. In a large bowl, beat the softened butter and granulated sugar until light and fluffy.

4. Beat in the eggs one at a time, then mix in the vanilla extract and lemon zest.

5. Add the dry ingredients alternately with the milk, mixing gently.

6. Add the fresh lemon juice and mix just until the batter is smooth. Do not overmix.

7. Pour the batter into the prepared pan and smooth the top.

8. Bake for 25 to 30 minutes, or until the center springs back lightly and a toothpick inserted in the center comes out clean.

9. While the cake bakes, combine the lemon juice, sugar, water, and lemon zest in a small saucepan.

10. Warm over medium heat, stirring until the sugar dissolves. Bring to a gentle simmer, then remove from the heat and stir in the vanilla.

11. Let the baked cake cool for about 5 minutes.

12. Use a skewer or toothpick to make small holes across the surface of the warm cake.

13. Slowly spoon the warm lemon syrup over the cake, allowing it to absorb gradually.

14. Cool the cake completely in the pan.

15. Lift the cake out using the parchment paper and cut into 16 squares.

16. Dust with powdered sugar and fresh lemon zest before serving.