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Heat the olive oil in a frying pan and fry the onions until softened but not coloured. Add the cumin and cook for 1 min, then scrape the mixture into a food processor with the chickpeas, two-thirds of the grated beetroot, the breadcrumbs, egg and tahini. Whizz to a rough paste, then scrape into a bowl and stir in the remaining grated beetroot with plenty of seasoning.
With damp hands, shape into about 20 balls and space on baking parchment-lined baking sheets. Chill until ready to serve.
Heat oven to 200C/180C fan/gas 6. Brush the falafels with a little vegetable oil and bake for 20-25 mins until crisp and hot through. Alternatively, heat 2.5cm oil in a deep wok and fry in batches, turning, until crisp and hot.
To serve, mix the tahini with the yogurt, sugar and some seasoning, then dollop onto the flatbreads with the warm or room-temperature falafels, diced beetroot and beet leaves or salad.