Preheat the oven to 350°F (175°C). Grease and flour an 8-inch (20 cm) round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, beat the softened butter and 1 cup granulated sugar together with an electric mixer 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 vanilla extract and lemon zest.
With the mixer on low speed, alternately add the flour mixture and the milk in three additions, beginning and ending with the flour mixture. Mix just until combined; do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 35-40 minutes, or until a toothpick inserted into the center comes out clean.
While the cake is baking, prepare the lemon syrup by combining 1/4 cup fresh lemon juice and 1/2 cup granulated sugar in a small saucepan over medium heat. Stir until the sugar dissolves completely and the syrup is clear. Remove from heat and set aside.
When the cake is done, remove it from the oven and let it cool in the pan for 10 minutes. Then, poke holes all over the top of the cake using a skewer or fork.
Slowly pour the warm lemon syrup evenly over the cake, allowing it to soak in. Let the cake cool completely in the pan on a wire rack.
Once cooled, prepare the lemon glaze by whisking together the powdered sugar, 2 tablespoons fresh lemon juice, and 1 teaspoon melted unsalted butter until smooth.
Remove the cake from the pan and place it on a serving plate. Drizzle the lemon glaze evenly over the top of the cake.
Allow the glaze to set before slicing and serving.