Heat oven to 180C/160C fan/gas 4. Grease and line a 20 x 29cm baking tray with baking parchment.
For the filling, zest the lemon half and leave aside for the dough. Peel, core and thinly slice the apples, then squeeze over the juice of the lemon to stop the fruit turning brown. Put the apples in a large pan and add the sugar, 200ml water and cinnamon. Cook for 5 mins, then remove from the heat and leave to cool in the liquid (you’ll need this later).
To make the dough, put the flour and baking powder in a food processor or into a large bowl and pulse or stir to combine. Add the butter and mix again until the mixture is sandy. Add the sugar, egg yolks and egg, yogurt, lemon zest and vanilla extract and mix into a dough. Tip it out onto a floured surface. Bring it together with your hands and roll it into a ball.
Split the dough in half, wrap one half in cling film and freeze for 1 hr. Roll out the other dough half so that it is big enough to fill the bottom of the lined tray. With the palm of your hand, push the dough about halfway up the sides of the tray until the whole base is covered. Prick the dough with a fork and bake in the oven for about 15 mins until it is golden and lightly springy to the touch.
Spoon over the apple filling, with about half the cooking liquid, then set aside.
Remove the dough from the freezer and coarsely grate, as you would a block of cheese. Sprinkle the grated dough over the apples and bake for 40-45 mins until it is golden and the topping has cooked through. Leave to cool completely, dust with icing sugar, then cut into squares. Whip the cream until thick, stir in the cinnamon and serve alongside the cake.