Preheat the oven to 350°F (175°C). Grease the 9-inch round cake pan and line the bottom with parchment paper.
In a mixing bowl, combine the flour, granulated sugar, baking powder, and salt. Mix well and set aside.
In a separate bowl, cream together the softened butter and remaining sugar until light and fluffy, using an electric mixer or whisk.
Add the eggs one at a time, beating well after each addition. Then mix in the lemon zest, lemon juice, and vanilla extract.
Gradually add the dry ingredients to the wet mixture, alternating with the milk. Start and end with the flour mixture. Mix until just combined, being careful not to overmix.
Pour the batter into the prepared cake pan and smooth the top with a rubber spatula.
Bake in the preheated oven for about 40-45 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, remove the cake from the oven and let it cool in the pan for about 10 minutes. Then, remove it from the pan, peel off the parchment paper, and transfer it to a cooling rack to cool completely.
Before serving, dust the top with powdered sugar for decoration.