Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a large mixing bowl, combine the flour, sugar, baking powder, and salt. Mix well to combine. Add the softened butter to the dry ingredients and mix until the mixture resembles fine crumbs.
In a separate bowl, whisk together the milk, eggs, lemon zest, lemon juice, and vanilla extract. Pour the wet ingredients into the dry ingredients and mix until the batter is smooth and well combined. Be careful not to overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula. Bake in the preheated oven for about 30-35 minutes, or until a toothpick inserted into the center comes out clean.
While the cake is baking, prepare the lemon glaze. In a small bowl, whisk together the powdered sugar and lemon juice until smooth. Once the cake is done, remove it from the oven and let it cool in the pan for 10 minutes. Then, transfer the cake to a wire rack.
While the cake is still warm, pour the lemon glaze over the top, allowing it to drizzle down the sides.
Let the cake cool completely before serving.