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Start by making the pastry case. Put the butter and flour in a food processor and pulse until it resembles fresh breadcrumbs. Add the yolk of the egg and the sugar and pulse again until mixed through then, add a tbsp at a time of very cold water pulsing the processor after each addition. Stop adding water as soon as the dough starts to come together as one piece.
Remove the dough from the processor and continue to bring together by kneading gently with your hands. Once you have a smooth ball of dough wrap it tightly in cling film and chill in the fridge for 30mins.
Heat oven to 190C/170C/gas 4 and grease a 23cm loose bottomed fluted tart tin. Roll out the chilled pastry on a lightly floured surface and use it to line the tin. Leave about half a cm of pastry proud of the edge of the tin - snipping it to length with kitchen scissors.
Line the base with a piece of baking parchment and weigh it down with baking beans or uncooked rice. Bake for 15 mins then remove the paper and beans and brush the pastry surface with beaten egg white. Return to the oven for a further 15-20 mins or until crisp, golden and cooked through. Leave to cool completely (can be done a day ahead).
Spread half of the caramel on the case of the pastry case and layer the sliced bananas on top. Cover with the rest of the caramel and place in the fridge.
Whip the cream until billowy thick and spoon it over the top of the tart. To serve finely grate over a 1-2 pieces of dark chocolate to garnish.