Preheat oven to 375 F. Grease and line an 8 x 8 inch square baking pan with parchment paper.
To make the dough, combine flour and baking powder in a food processor or a large bowl and stir to combine. Add the butter and pulse in the food processor, or mix with a pastry cutter until the mixture is sandy. Add the sugar, egg yolk and egg, yogurt, lemon zest, and vanilla extract. Mix together. Tip onto a flour surface and knead gently until a ball forms.
Split dough in half and wrap one half in plastic wrap and freeze for 1 hour. Put the remaining dough half in the baking pan and use the palm of your hand and your fingertips to evenly distribute into a uniform dough layer across the bottom of the baking pan and halfway up the sides of the pan. Prick the dough with a fork and bake in the oven for about 15 minutes until it is golden and lightly springy to touch. Set aside to cool.
For the filling, peel, core, and thinly slice the apples. Pour the lemon juice over the apple slices in a medium bowl. Place the sliced apples in a large pan and add sugar, 100 ml water, and cinnamon. Cook for 5 mins over medium heat then remove from heat and leave to cool in the liquid.
Spoon the apple mixture over the base, including about half of the cooking liquid. Set aside. You should have a thick layer of apples.
Remove the dough from the freezer and coarsely grate the dough over the top of the apple layer.
Bake 45-50 minutes until the topping is golden and cooked through.
Leave to cool fully, then dust with icing sugar and cut into squares.
Serve slightly warmed topped with a scoop of vanilla ice cream.