Preheat the oven to 200C/400F/Gas 6.
For the pastry, pulse the flours and butter in a food processor until the mixture resembles fine breadcrumbs. With the motor running, add the egg in a thin stream and blend until the mixture begins to form a ball.
Roll the pastry into a circle on a floured work surface and use it to line a 23cm/9in loose-based fluted tart tin. Press the pastry with your fingertips firmly into the base and sides. Trim any excess pastry and lightly prick the base of the tart using a fork. Chill in the fridge for 30 minutes.
Line the pastry with a large sheet of crumpled baking parchment and half fill with baking beans. Bake the tart case for 25 minutes. Remove the beans and return the tart to the oven for a further 5-10 minutes or until the surface is dry and beginning to brown. Remove from the oven and reduce the oven to 180C/350F/Gas 4.
For the filling, melt the butter in a large heavy-based frying pan over a low heat. Add the leeks and fry gently for 2-3 minutes, stirring until just softened. Remove from the heat and set aside.
Put the eggs in a large jug and beat lightly with a whisk. Stir in the crÃ¨me fraÃ®che, season with salt and freshly ground black pepper. Beat together with a wooden spoon until thoroughly combined. Stir in the brown crab meat.
Scatter the leeks over the pastry case and dot the white crab meat around them. Pour over the crÃ¨me friache mixture and sprinkle with the cheese. Cook on the baking tray for 25 minutes, or until the filling is golden-brown and just set. (It should still wobble a tiny bit in the centre as it will continue to set as it cools.)
Remove the quiche from the oven and leave to cool in the tin for 15 minutes before removing.