1. Preheat the oven to 350°F. Grease and line a 9 x 12-inch or 9 x 13-inch cake pan with parchment paper.

To Make the Apple Filling

1. Zest the lemon half and set it aside so it can be used for the dough.

2. Peel, core and thinly slice the apples, tossing the slices with a little lemon juice as you work to prevent the fruit from turning brown. Place the apples in a large skillet and add the brown sugar, water and cinnamon. Cook for 5 minutes over medium heat until just tender. Remove from the heat and leave to cool in the pan.

To Make the Dough

1. Put the flour, baking powder and butter into the bowl of a large food processor. Pulse until the mixture is sandy in texture. Add the lemon zest, sugar, egg yolks, whole egg, yogurt and vanilla extract. Pulse until the mixture transforms into a dough.

2. Tip the dough out onto a floured surface. Bring it together with your hands and roll it into a ball. Divide the dough into two fairly equal pieces. Wrap one piece in plastic wrap and place it in the freezer for 20 minutes.

To Assemble the Cake

1. On a floured surface, roll out the remaining dough  so that it is large enough to fill the bottom of the lined cake pan. Press the surface of the dough to be sure that the whole base is covered and the dough reaches all four corners.

2. Prick the dough all over with a fork. Bake in the oven for about 15 mins until it is golden and lightly springy to the touch.

3. Drain most of the liquid from the cooled apple mixture. Spoon the apple filling evenly over the surface of the baked dough.

4. Remove the dough from the freezer and coarsely grate it, as you would grate cheese. Sprinkle the grated dough evenly over the apples. Return the pan to the oven for 45 minutes until the top of the cake is nicely golden.

5. Leave the cake to cool in the pan. Dust with confectioner's sugar and cut into squares or bars.