Preheat the oven to 220°C/gas mark 7. Wrap each of the beetroots in tinfoil and roast for 1 hour, until soft when inserted with a skewer. Leave to cool slightly then peel off the skin – this should come away easily. Lower the oven to 140°C/gas mark 1.
Meanwhile, combine the ingredients for the dipping sauce. Season to taste and set aside in the fridge.
Place the beetroot in a food processor and whizz until fairly smooth. Whisk together the eggs and milk. Place in a large bowl and add the beetroot, then stir together until smooth. Fold through the feta, dill, flour and baking powder and season.
Heat the oil in a wide frying pan over a medium heat until a small drop of the mixture sizzles when added to the pan. Add 5-6 generous tablespoons of the mixture to the hot oil and press each of these with the back of a spoon slightly so they are flat, making sure they do not touch each other. Cook for 3-4 minutes on each side, until browned and slightly puffed up. Lower the heat if they start to brown too quickly. Transfer to a baking sheet lined with kitchen paper to absorb excess grease and keep warm in the low oven while you cook the rest. Cook in batches until all the mixture is used up. You should end up with 24 fritters.
Serve 6 fritters each with the dipping sauce, garnished with the extra dill.