Preheat the oven to 160 C. Grease a 9 inch round springform tin and set aside.
In the bowl of a stand mixer, combine the sugar and lemon zest. Rub the zest into the sugar with your fingertips until the sugar is fragrant and slightly moistened. Now add the eggs and vanilla.
Use the whisk attachment and beat for 3 to 4 minutes on high speed until very thick and light in colour. When you lift up the whisk, the batter should fall in ribbons back into the bowl. If this doesn't happen, continue beating because this step is crucial to the perfect texture. You can also use a hand mixer but make sure the bowl is very large and deep.
Sift in the flour, baking powder and salt. Pour in the lemon juice. Switch to the paddle attachment if using a stand mixer and beat on low speed until just combined.
Now in a small saucepan, heat the milk until it's hot but not boiling. Take it off the heat, then add the oil and butter to the milk, stirring until the butter has dissolved.
With the mixer running on low speed, pour this hot milk mixture in a thin stream into the bowl. If using a hand mixer, an extra set of hands is useful at this step so that you can continue beating the batter, otherwise pour it in gradually, alternating with beating the batter.
Give the batter a quick stir by hand to make sure it's evenly mixed and there's no bits of flour stuck to the bottom. It is a fairly runny batter.
Pour into the prepared tin and smoothen for top. Bake for 50 to 55 minutes until lightly browned on top and a toothpick poked in the center comes out clean. The top might dome a bit, but will be levelled off before frosting.
Allow to cool completely at room temperature.
To make the frosting, beat the butter and cream cheese with a hand mixer on low medium speed, until smooth and creamy. Add the lemon zest and beat briefly to combine, then add the icing sugar. Beat again until the sugar has incorporated, then add the cream and beat again till smooth. Place it in the fridge for about 20 minutes before frosting the cake. The frosting isn't super thick, but if you prefer to add more sugar to thicken it, you can.
Slice the cake horizontally in half using a long, sharp knife. Lift off the top half carefully and place it on a tray. Spread half the frosting over the lower half, then place the top half back on it, pressing down lightly.
Spread the rest of the frosting on the top and let it spill over the sides. Smoothen the sides to give it a 'naked look'. Refrigerate the cake for 30 minutes, then decorate with more lemon zest and dig in!
Enjoy immediately or store in the fridge in an airtight tin for 5 days. Best eaten at room temperature so that the frosting softens. The cake also freezes and thaws beautifully. Happy baking!