Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, beat the softened butter and granulated sugar together on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the lemon zest, lemon juice, and vanilla extract until combined.
Gradually add the flour mixture to the wet ingredients alternately with the milk, beginning and ending with the flour mixture. Mix on low speed just until combined; do not overmix.
Pour the batter into the prepared baking pan and spread evenly.
Bake in the preheated oven for 28 to 30 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and allow the cake to cool completely in the pan on a wire rack.
Once cooled, dust the top of the cake with powdered sugar.
Cut into 8 equal slices and serve.