Preheat the oven to 220°C / 425°F / Gas Mark 7. Lightly grease six 11cm straight-sided cake or tart tins and then line with baking parchment.
To make the frozen yogurt, put the yogurt, icing sugar and vanilla in a 15 x 22cm igid plastic container and mix well.
Level it with the back of a spoon, then cover and pop into the freezer for about 8 hours or overnight until set firm.
This will need to be given a good stir two to three times in this time period to help avoid any ice crystals forming. This can obviously be prepared well in advance.
Peel the apples. Then with one of them sitting stem end up on a chopping board, slice off one of th 'cheeks' on one side (as close to the core as possible). Then slice off the oposite cheek and discard the middle bit (o nibble around it for a chef's perk).
Repeat with the remaining apples. Toss them in a small bowl with the cinnamon and ginger and leave aside for now.
Put the butter, sugar and 4 tbsp cold water together in large non-stick frying pan over a low to medium heat.
Stir until the butter melts and the sugar dissolves and then continue to cook for about 8 minutes, swirling the pan occassionally.
The sugar will caramelise, turning from light brown to a medium brown.
Now add the apple cheeks, cut-side down, and allow to cook for 2 minutes.
Baste with the caramel, turn the apples over and cook for a further 3 minutes until just beginning to soften.
Take the pan off the heat and leave the apples to sit in there and cool down to body temperature while you prepare the tins and pastry.
Roll out the puff pastry on a lightly floured surface to about a 26 x 40cm rectangle.
Put a tart tin upside down near one corner of the rectangle and cut around it.
Repeat to give six pasry discs in total and then prick them all over with a fork.
Once the apples have cooled down, remove them from the caramel onto a plate.
Divide the caramel into the bottom of each tart tin. Put one apple cheek, cut-side up, on the sauce in each tin.
Place a pastry disc on top, tucking it down the sides of the apple so the pastry is nice and tight all the way around.
Cover with cling film and pop in the fridge for 10 minutes or the freezer for 5 minutes to firm up a bit. This is so the pastry gets a nice rise when cooking and the flour in the pastry cooks before the butter in the pastry melts to a gooey mess. The tarts can be prepared to this stage up to a day ahead of time.
After this time, bake the tarts in the oven for 15-20 minutes or until the pastry is firm where risen and a golden brown.
Remove the frozen yogurt from the freezer to allow to come to room temperature.
Meanwhile, roughly chop the pecans and place them in a small frying pan (with no oil in) over a medium heat. Cook for 3 minutes or so, tossing occassionally until you start to smell the aroma coming from the nuts. Then tip these out onto the plate.
Once cooked, invert a serving plate on top of a tarte tatin and then flip both over. Remove the tin to reveal the tart and allow the caramel to flow out. Repeat with the remaining tarts.
Scatter over the pecans, a little sprinkling of ground ginger and decoate each with a mint sprig, if liked.
Serve straight away with a scoop of frozen on top.