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 mixing bowl, cream the softened unsalted butter and granulated sugar together using 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 to fully incorporate.
Mix in the lemon zest and vanilla extract until evenly combined.
Reduce the mixer speed to low and alternately add the dry flour mixture and whole milk to the butter mixture, beginning and ending with the flour mixture. Mix until just combined; do not overmix.
Stir in the 2 tablespoons of fresh lemon juice gently with a spatula.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 40-45 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 10 minutes.
While the cake is still warm, prepare the lemon drizzle by mixing the powdered sugar, 2 tablespoons fresh lemon juice, and 1 teaspoon lemon zest 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.
Let the cake cool completely in the pan on a wire rack before removing and serving.