Preheat the oven to 350°F (175°C). Grease a 9 x 13-inch baking pan and lightly flour it, or line it with parchment paper.
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 on medium speed until light and fluffy, about 3 to 4 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the vanilla extract, lemon juice, and lemon zest until combined.
Alternately add the dry flour mixture and the milk to the butter mixture, beginning and ending with the flour mixture. Mix on low speed just until combined after each addition. Do not overmix.
Pour the batter evenly into the prepared 9 x 13-inch pan and smooth the top with a spatula.
Bake in the preheated oven for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan on a wire rack for 15 minutes.
While the cake cools, prepare the lemon glaze by whisking together the powdered sugar and 2 tablespoons of fresh lemon juice until smooth.
After the cake has cooled for 15 minutes, drizzle the lemon glaze evenly over the top of the cake.
Allow the cake to cool completely before slicing and serving.