Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper for easy removal.
In a mixing bowl, cream together the softened butter and granulated sugar until light and fluffy using an electric mixer or whisk.
Add the eggs one at a time, beating well after each addition. Mix in the lemon juice, lemon zest, and vanilla extract until combined.
In another bowl, sift together the all-purpose flour, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, alternating with the milk. Start and end with the flour mixture, mixing just until combined. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 25 to 30 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let the cake cool in the pan for 10 minutes, then transfer to a cooling rack to cool completely.
For the lemon glaze, combine the powdered sugar, lemon juice, and lemon zest in a small saucepan over low heat. Stir until smooth and slightly warmed.
Drizzle the glaze over the cooled cake, allowing it to drip down the sides. Let it set for a few minutes before serving.