Preheat the oven to 200C/400F/Gas 6. Grease a 28cm/11in tart case with butter.
For the sweet pastry, in a food processor, pulse the butter and flour until the mixture resembles breadcrumbs.
Transfer the butter and flour mixture to a bowl and add the icing sugar and vanilla seeds. Mix until well combined.
Add the beaten egg and mix, using your hands, until the mixture comes together as a dough. Wrap the dough in cling film and chill in the fridge for 30 minutes.
Roll the chilled pastry out onto a lightly floured surface until it is 30-35cm/12-14in in diameter. Line the prepared tart tin with the pastry, pressing it gently to the tin and trimming off any excess. Line the pastry case with baking parchment and half-fill with ceramic baking beans, then transfer to the oven and bake for 8-10 minutes.
Remove the baking beans and baking parchment, then prick the pastry several times with a fork. Return to the oven and continue to cook for a further 4-5 minutes, or until the pastry is crisp and pale golden-brown.
Reduce the oven temperature to 130C/250F/Gas 1.
For the tart filling, bring some water to the boil in a pan, then reduce the heat until the water is simmering. Suspend a heatproof bowl over the water. Do not allow the base of the bowl to touch the water. Add the dark chocolate pieces to the bowl and stir until smooth and completely melted. Remove the bowl from the heat and set aside.
Bring the milk and cream to the boil in a pan, then pour the mixture over the melted chocolate.
Add the beaten eggs and mix until smooth and well combined.
Pour the filling mixture into the pastry case and bake for 20-25 minutes, or until set. Allow to cool before serving.
To serve, place one slice of tart into the centre of each serving plate. Spoon a little clotted cream alongside.