First make the pastry. Mix the flour with the sugar and butter in a large bowl. Use your fingertips to mix the ingredients until they resemble breadcrumbs. Stir in the eggs until it becomes a smooth paste then wrap the pastry in Cling film and place in the fridge for at least 3 hours, but preferably overnight.
Roll the dough on a floured surface and line each of the individual tartlet tins with the pastry (you will need 9 inch loose bottomed tins, greased and floured) Then chill these in the fridge for another 30 minutes. When ready to bake place some baking paper on the pastry and fill with rice grains or dried beans. Bake ‘blind’ in a hot oven for 10-15 minutes until the pastry has a pale golden colour. Remove the paper and the baking beans and seal the pastry case by brushing one slightly beaten egg white over the pastry. Put back in the oven and bake for a further 3 minutes then remove from the oven and leave to cool on a wire rack.
For the filling, mix the juice and zest of the lemons, cream, eggs and sugar in a bowl until fully incorporated and pour the mix into the pastry case.Turn the oven temperature down to 130ºC and bake for 45 minutes until the centre is just set. Remove from the oven and leave to cool.
Purée the two punnets of blackberries and set aside. To make the ice cream, split the vanilla pod in half and scoop the seeds into a pot along with the pod, the milk and the cream, bring everything to just under the boil. Whisk the egg yolks and sugar in a large bowl until they are pale and thick. Pour the hot cream and milk mixture over them, whisking all the time. Sieve this mixture back into the pot and cook on a low heat until the mixture coats the back of a spoon, stirring constantly. Remove from the heat and fold in the puréed blackberries until fully combined. Churn in an ice cream machine until thick.
To serve, place a lemon tartlet on a plate and spoon a scoop of the blackberry ice cream beside it and some whipped cream.