Preheat the oven to 150C/300F/Gas2.
Extract all the meat from the crab but reserve all the shells. Place the shells with the olive oil in a shallow roasting pan with the carrot, leek and onion. Bake in the oven for 2 hours to gently extract the flavour. Leave to cool then strain the oil and discard the debris.
Cook the potatoes in lightly salted boiling water until soft. Mash with a fork and mix in three-quarters of the white crab meat, half the brown crab meat and one third of the crab oil. Correct the seasoning with salt. Divide the mixture into balls each weighing about 50g/2oz. On a floured surface, shape into round flat patties with a palette knife. Pass the cakes first through the flour, then the egg and then the breadcrumbs.
Put the remaining crab oil, the tomatoes, sliced basil, lemon juice and the rest of the crab meat in a pan and warm through but do not boil.
Set the deep-fryer to 160C/320F or heat some oil to the same temperature in a large pan. Add the cakes and fry for three minutes. Remove and drain on kitchen paper.
Divide the crabcakes between four warmed plates. Pour the dressing around the crab cakes. Garnish with a whole basil leaf placed on top of the dressing. Serve immediately.