Special equipment: six 4 1/2-inch straight-sided cake or tart pans
For the frozen yogurt: Put the yogurt, confectioners' sugar and vanilla in a 6-by-8 1/2-inch rigid plastic container and mix together well. Level it with the back of a spoon.
Cover and pop into the freezer for about 8 hours or overnight ¿ the yogurt should be set firm. The yogurt will need to be given a good stir 2 to 3 times during freezing to help avoid any ice crystals forming. This can obviously be prepared well in advance.
For the tarte tatin: Peel the apples. With one of them sitting stem-end up on a chopping board, slice off one of the 'cheeks' on one side (as close to the core as possible). Slice off the opposite cheek and discard the middle bit (or 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 set aside for now.
Put the butter, superfine sugar and 4 tablespoons cold water in large non-stick frying pan over 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 occasionally. The sugar will caramelize turning from light brown to a medium brown.
Add the apple cheeks, cut-side down, and cook for 2 minutes. Baste with the caramel, turn the apples over and cook for a further 3 minutes, or until just beginning to soften. Take the pan off the heat and leave the apples to sit and cool down to body temperature while you prepare the pans and pastry.
Preheat the oven to 425 degrees F. Lightly grease six 4 1/2-inch straight-sided cake or tart pans and then line with baking parchment. Arrange on a baking sheet and set aside.
On a lightly floured surface, roll out the puff pastry to a rectangle large enough to cut out 6 discs to fit the tart pans. Put a tart pan upside down near one corner of the pastry and cut around it. Repeat to give 6 pastry discs in total and then poke 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 bottoms of each tart pan. Put one apple cheek, cut-side up, on the sauce in each pan. Place a pastry disc on top, tucking it down the sides of the apple so that the pastry is nice and tight all the way around.
Cover with plastic wrap and pop in the fridge for 10 minutes (or the freezer for 5 minutes) to firm up a bit. This is so that 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 1 day ahead of time.
After this time, bake the tarts in the oven until the pastry is firm where risen and golden brown, 15 to 20 minutes. Remove the frozen yogurt from the freezer to allow to come to room temperature.
Meanwhile, for the garnish: Roughly chop the pecans and place them in a small frying pan (with no oil) over medium heat. Cook for 3 minutes or so, tossing occasionally, until aromatic. Tip the nuts onto a plate.
Once cooked, place a serving plate on top of one tarte tatin and then flip both over. Remove the pan to reveal the tart and allow the caramel to flow out. Repeat with the remaining tarts.
Scatter over the pecans and a little sprinkling of ground ginger, and decorate each with a mint sprig, if desired. Serve straightaway with a scoop of frozen yogurt on top.