Peel the potatoes and slice very finely - a mandolin is the best way to do this. Cut the beetroot into fine slices as well - they don't have to be as thin. You can cut them with a knife.
In a large saucepan, mix together the creams and the milk and bring the liquid to just under the boil. Add the sliced potatoes and cook gently for 5 minutes.
Season really well and spoon half the potatoes into a buttered gratin dish. Put a layer of beetroot on top - seasoning as you go, then top with the rest of the potatoes and their cream. Put in the oven for 1 hour, or until the vegetables are completely tender. You may need to cover the top with foil after 45 minutes to stop the top becoming too dark in color.