In a large bowl, sift together the flour, cocoa and bicarbonate of soda, then stir in the sugar. In a food processor, purée the beetroot, then scrape it into a sieve set over a bowl and push out the juices with the back of a spoon. Set them aside for the icing. Tip the beetroot pulp back into the food processor, then, with the motor running, add the eggs and vanilla, then slowly pour in the oil. Mix until blended.
Make a well in the centre of the dry ingredients, pour in the beetroot mixture and, with a large spoon, gently fold together. Pour into the tin and
Remove the cake from the oven, leave it for 5-10 minutes in the tin, then turn it out on to a wire rack until completely cold.
Sift the icing sugar into a bowl. Stir in a few drops of the reserved beetroot juice and, depending on how bright you want the colour, a little pink food colour (if you like). The icing should be thick enough to coat the back of a spoon. If not, add water, drop by drop, until you achieve the correct consistency.
Turn the cake over, so the base becomes a flat top, and spread the icing evenly over, letting it drizzle down the sides.