Preheat the oven to 350°F (175°C). Grease and line an 8×8 inch (20×20 cm) square baking pan with parchment paper.
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 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the lemon zest and vanilla extract until combined.
Alternately add the dry flour mixture and the milk to the butter mixture, beginning and ending with the flour mixture. Mix just until combined.
Stir in the fresh lemon juice gently until incorporated.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake for 28 to 30 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 cools, prepare the lemon glaze by whisking together the powdered sugar and fresh lemon juice in a small bowl until smooth.
After 10 minutes, remove the cake from the pan using the parchment paper and place it on a wire rack set over a baking sheet.
Pour the lemon glaze evenly over the warm cake, allowing it to drip down the sides.
Let the cake cool completely until the glaze is set, about 30 minutes.
Cut into 12 equal slices and serve.