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 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 milk to the butter mixture, beginning and ending with the flour mixture. Mix until just combined.
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 of the cake comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes. Then transfer the cake to a wire rack to cool completely.
While the cake is cooling, prepare the lemon drizzle by whisking together the powdered sugar and freshly squeezed lemon juice until smooth.
Once the cake is completely cool, drizzle the lemon glaze evenly over the top, allowing it to drip down the sides.
Let the drizzle set for about 15 minutes before slicing and serving.