Preheat the oven to 160 C. Line 10 muffin moulds with liners and set aside.
In the bowl of a stand mixer fitted with the whisk attachment (or use a mixing bowl and hand mixer), combine the eggs, sugar and vanilla. 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.
Sift in the flour, baking powder and salt. 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. Turn 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. Don't worry, the eggs will not scramble! 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.
Divide equally into the cupcake liners. It's easiest to just pour the batter rather than spoon it in.
Bake for 20 to 25 minutes until a toothpick poked in the center comes out clean. The cupcakes don't brown or dome much, but the tops will feel springy when they're done.
Allow to cool completely at room temperature.
To make the frosting, sift the cocoa and icing sugar together in a small bowl. Add the hot water and stir well to make a smooth, thick paste-like mixture. I found that the hot water creates a silkier frosting although I've made similar frostings without it as well.
Now in a mixing bowl, beat the butter, vanilla and the cocoa mixture, until you have a smooth, fluffy and lightened frosting. If it feels runny, chill it for 15 minutes.
Spoon or pipe onto the cooled cupcakes and decorate with sprinkles if you like. Enjoy immediately or store in the fridge in an airtight tin for 5 days. Best eaten at room temperature so that frosting softens. Happy baking!