Preheat the oven to 350°F.
Grease a good cake pan well, and line it with a parchment circle to fit the bottom of the pan to prevent the cake from sticking.
Add flour, sugar, cocoa powder, and baking soda, and a pinch salt to a bowl of a food processor. Pulse a few times to incorporate ingredients and remove lumps if any.
In a large measuring jug, add water, olive oil, lemon juice, and vanilla and mix well
Add liquid ingredients to the dry ingredients in the food processor bowl.
Process to get a smooth batter. It does not take long, don't over mix.
Tranfer the batter to the prepared pan, bake in the preheated oven for 30 minutes.
Remove from the oven and let it cool competely on a rack.
Meanwhile, add the frosting ingredients; such as, softened butter, powdered sugar, cocoa, and vanilla to a large, deeper bowl to prevent powdered sugar flying around.
Use a hand held mixer to combine the ingredients, and start adding milk in ½ teapoon portions at a time until you get the desired frosting consistency.
If you dont want to use butter, increase the amount of milk. frosting takes very little liquid, so be careful. Otherwise it will turn to a glaze, which can be used but not as frosting.