Preheat the oven to 350 F (175 C). Line a round 9-inch cake tin with parchment paper or butter and flour so that the cake doesn't stick.
Cream together the butter and sugar for about 5 minutes until very light and fluffy. Make sure to scrape down the sides a few times.
Add the eggs one at a time, letting the previous one become fully incorporated before adding the next. Make sure they are room temperature, so that your batter doesn’t split!
Add the vanilla extract and lemon zest to the eggs, butter, and sugar. Mix together.
Measure out the lemon juice in a separate cup or bowl. Mix the dry ingredients - flour, baking powder, and salt - together in a separate bowl.
Sift in half of the dry ingredients and mix together until just incorporated. Then add the lemon juice and mix until incorporated. Sift in the remaining dry ingredients and mix.
Add the milk. Once incorporated, give the mixture a few folds with a spatula to make sure everything from the bottom has been mixed.
Pour the batter into the cake tin.
Bake for about 30-35 minutes or until a skewer/knife inserted into the center of the cake comes out clean. Note: if you take the cake out early, it may collapse!
To mix the icing, just combine powdered sugar with water. Start by adding a couple of drops of water to a small bowl of powdered sugar. Continue to add water or powdered sugar until you get the consistency that you want. Keep in mind that a thicker icing will be difficult to spread on the cake without ruining the top, and a thinner icing will just run right off. So, you want an icing with a consistency somewhere in the middle!