Whiz the beetroot in a food processor until fine. Set aside in a bowl.
In the food processor, pulse the flour, butter, icing sugar and salt until they resemble fine breadcrumbs.
Add the beetroot and pulse to combine. Add the cold water, a tablespoon at a time until the dough comes together in a ball. Don't get it too wet.
Roll the dough out on a lightly floured surface and then line the greased pie dish with the pastry. Gently press it into the flutes and trim the edges by rolling the rolling pin over the edge. Prick the bottom with a fork and refrigerate the pastry case for at least half an hour.
Preheat the oven to 180C/350F and line the bottom of the pastry case with parchment and cover the base with baking beans (or uncooked rice). Bake for 12 minutes then remove the beans and parchment and cook for another 7 minutes or until just starting to turn golden. Allow to cool.
For the filling:
Melt the white chocolate and allow to cool slightly.
Sift the icing sugar and arrowroot into the yogurt and vanilla and beat together. Add the chocolate and then stir into the mixture.
Pour into the cooled pastry shell and refrigerate for a few hours or overnight to firm up. Serve cold with finely grated white chocolate on top to decorate.