Preheat the oven to 350°F (175°C). Grease and flour an 8-inch (20 cm) round cake pan or line it with parchment paper.
In a medium bowl, whisk together the sifted all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, cream the softened unsalted 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 to fully incorporate.
Mix in the lemon zest and vanilla extract until evenly combined.
With the mixer on low speed, alternately add the dry flour mixture and the milk in three additions, beginning and ending with the flour mixture. Mix just until combined; do not overmix.
Gently fold in the fresh lemon juice with a spatula until evenly distributed.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 38-40 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
While the cake is still warm, prepare the lemon drizzle by mixing the powdered sugar and fresh lemon juice in a small bowl until smooth.
Prick the top of the cake all over with a skewer or fork, then slowly pour the lemon drizzle evenly over the warm cake, allowing it to soak in.
Allow the cake to cool completely in the pan on a wire rack before removing and serving.