Preheat the oven to 180C/160C Fan/Gas 4. Grease and line a 30x20cm/12x8in traybake tin.
Mix together the flour, cocoa powder and baking powder in a bowl.
Blend the beetroot in a food processor until you have a rough purée. Put the purée in a bowl, add the vinegar and buttermilk. Mix to combine.
Beat the butter and sugar together until light and fluffy. Gradually beat in the eggs, followed by half the dry mixture and half the beetroot mixture.
Continue to mix gently, adding the rest of the dry ingredients and the remaining beetroot. Pour the mixture into the tin and bake for 40-50 minutes. leave to cool in the tin for 10 minutes before turning it out of the tin and leaving it to cool completely.
Meanwhile make the icing. Beat together the cream cheese and butter until soft. Beat in the icing sugar and vanilla extract until they are combined, then spread the icing onto the cooled cake.
For the beetroot hearts, thinly slice the beetroot, and cut into 12 heart shapes.
Boil the sugar in 200ml/7fl oz water until the sugar has dissolved, then add the beetroot hearts and simmer for about 10 minutes until soft and translucent. Using a fork, carefully remove the beetroot hearts from the hot sugar syrup and carefully lay them on paper towel to soak up any excess syrup.
Slice the cake into 12 even squares, and then decorate each square with a beetroot heart.