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, cream the softened butter and granulated sugar together 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 dry ingredients to the wet ingredients alternately with the milk, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake for 28 to 30 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and let the cake cool completely in the pan on a wire rack.
Once cooled, cut the cake into 8 slices and dust with powdered sugar if desired before serving.