Preheat the oven to 120°C/250°F, and line a roasting pan with baking paper.
Add the pears, sugar, and vanilla (seeds and pod) to a bowl and mix to combine. Place the pears (cut side up) into the roasting pan.
Top the pears with the pieces of butter and roast in the oven for 2 hours or until the pears are tender. Check every half hour or so to turn the pears and baste them in the juices.
Pâte Sucrée
In a food processor, combine the flour, icing sugar, and salt until well-combined.
Add the butter and iced water and process until the mixture resembles fine crumbs. Then add the egg yolk. Process to combine.
Turn out the pastry onto a cling film and form into a disc using the back of your hand. Wrap dough in cling film and refrigerate for 2 hours.
Grab 1 large, 2 medium, or 8 mini fluted tart tins.
Retrieve the pastry from the fridge. Roll out the pastry between 2 sheets of lightly floured baking paper to a thickness of 2mm. Line the tart tins with the pastry and trim the edges. Refrigerate for 30 minutes.
Preheat the oven to 180°C/355°F.
Line the tart crusts with baking paper, add pastry weights, and blind bake the tarts until golden—about 15 minutes. Remove the baking paper and pastry weights, and brush the inside of the tart with the beaten egg. Bake again until crisp—about 5 minutes. Set aside to cool.
Once cool, thinly slice 3 pear halves from the roasting pan. Line the tart bases with the halved pears.
Pedro Ximénez Chocolate Ganache
Place the cream and Pedro Ximénez in a small saucepan and bring to a simmer over a medium heat.
Remove from the heat and add the chocolate, stirring until smooth. Let chocolate mixture stand for 10 minutes before pouring into the tart shells over the pear slices.