Preheat the oven to 350°F (175°C). Grease and flour a 9 x 13-inch 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 3-4 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the lemon zest, fresh lemon juice, and vanilla extract until combined.
With the mixer on low speed, alternately add the flour mixture and the milk to the butter mixture, beginning and ending with the flour mixture. Mix just until combined; do not overmix.
Pour the batter evenly into the prepared 9 x 13-inch pan and smooth the top with a spatula.
Bake for 33 to 37 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 for 15 minutes.
While the cake is cooling, prepare the lemon glaze by whisking together the powdered sugar and fresh lemon juice in a small bowl until smooth.
After the cake has cooled for 15 minutes, drizzle the lemon glaze evenly over the top.
Allow the cake to cool completely before slicing and serving.