Heat oven to 180C/160C fan/gas 4. Grease and line a deep 20cm springform cake tin with baking parchment. Tip the beetroot into a food processor and whizz to a purée. Add the remaining ingredients, along with ¼ tsp salt, and blend until well combined. Scrape into the cake tin, level the surface and bake for 50 mins-1 hr or until a skewer comes out clean.
Leave the cake to cool in the tin while you make the icing. Put the ingredients in a small saucepan, heat and whisk until smooth. Cool for 20 mins.
Flip the cake onto a wire rack, flat-side up. Pour over the icing and leave to cool completely. Sprinkle with dark chocolate shavings (if using), then serve.