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Halve 1 of the aubergines, from stalk to bottom, then cut into long, thin slices. Brush with 1 tbsp of the oil, then barbecue or griddle in batches until soft and charred. Set aside. Meanwhile, put the remaining 2 aubergines directly on the barbecue, or under a hot grill, and cook, turning regularly, until the skin is blistered and blackened and the flesh is really soft. Leave to cool.
Halve the whole aubergines, scoop out the flesh into a food processor and discard the blackened skins. Add the garlic, lemon zest and juice, tahini and seasoning, then whizz until smooth. Chill in the fridge and remove 30 mins before serving.
Heat oven to 200C/180C fan/gas 6. Brush the flatbreads with the remaining oil, place on a baking tray and bake for 10 mins until crisp.
Spread the flatbreads with the aubergine purée, then top with the griddled aubergine slices, pomegranate seeds, pine nuts and mint leaves. Cut into slices before serving.