First make the pastry. Rub the flour, sugar and butter together until they resemble breadcrumbs. Lightly beat the whole egg, split the vanilla pod in half lengthways and scrape out the seeds. Stir the egg and vanilla seeds into the flour and butter mixture, then knead it gently to form a dough. Shape into a disc, then cover with clingfilm and chill in the fridge for 30 minutes.
Roll the pastry out on a lightly floured work surface into a 3mm thick circle, about 30cm in diameter. Use the rolling pin to transfer it to a 24cm diameter, 6cm deep tart tin. Line the tin with the pastry, pressing carefully into the edges, making sure it hangs at least 2cm over the rim. Chill in the fridge for 30 minutes. Preheat the oven to 180C/350F/Gas mark 4.
Line the pastry case with greaseproof paper and fill it with baking beans, rice or flour. Put the tin on a baking tray and bake for 15 minutes. Remove the paper and contents and bake for a further 5 minutes. Beat the egg yolk. Carefully trim off the overhanging pastry edge level with the top of the tin, then brush the inside of the case with the egg yolk to seal the pastry. Put it back in the oven for a further 5 minutes, then remove from the oven.
Reduce the temperature to 150C/300F/Gas mark 2. Peel the bananas and put them in the base of the tart case to fill the bottom. In a bowl, beat together the eggs, egg yolks and sugar. Split the vanilla pod in half lengthways and scrape out the seeds. Bring the cream and vanilla to the boil in a small pan. Once boiling, whisk the cream into the egg mixture, beating all the time. Pour this into the pastry case over the bananas and bake for 1 1/2 hours, until the mixture has set and is just golden brown.