Preheat your oven to 350°F (175°C). Grease and flour a 9×13-inch baking pan or two 9-inch round cake pans.
In a large mixing bowl, cream together the softened butter and granulated sugar until light and fluffy.
Add the eggs one at a time, beating well after each addition.
In a separate bowl, whisk together the flour, baking powder, and salt.
Gradually add the dry ingredients to the butter mixture, alternating with the milk, beginning and ending with the dry ingredients. Mix until just combined.
Stir in the vanilla extract, lemon zest, and lemon juice until evenly incorporated into the batter.
Pour the batter into the prepared baking pan(s) and spread it evenly.
Bake in the preheated oven for 30-35 minutes (for the 9×13-inch pan) or 25-30 minutes (for round pans), or until a toothpick inserted into the center comes out clean.
While the cake is baking, prepare the glaze by whisking together the powdered sugar, lemon juice, and lemon zest until smooth.
Once the cake is done baking, remove it from the oven and let it cool in the pan for about 10 minutes.
Poke holes all over the top of the cake using a skewer or fork, then pour the glaze evenly over the warm cake.
Allow the cake to cool completely before slicing and serving.