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Grease and line the base and sides of a deep 20cm round cake tin with baking parchment, leaving 2.5cm of parchment sticking up above the rim, like a collar. In a large pan, melt the butter with the sugar. Mix together the cinnamon and biscuit crumbs, then stir into the butter and mix until it looks like wet sand. Press the crumbs into the base in an even layer, then freeze.
Put the apple, lemon juice and 1 tbsp water into a large pan. Bring to a simmer, then cover and cook for 3 mins or until totally soft. Pass this pulpy mix through a sieve into a large bowl. Leave to cool. Mash the blackberries with a fork, then sieve into another bowl – a rubber spatula is best for pressing every last bit of fruit from the seeds.
Stir together the condensed milk, 500g yogurt and 300ml cream until evenly mixed. Pour 450g of this onto the apple purée, add 1⁄4 tsp cinnamon, then whip with a balloon whisk until slightly thickened. Pour onto the biscuit layer, level, then freeze for 30 mins or until firm. Chill the remaining yogurt mixture in the fridge.
Whisk about 3 tbsp of the blackberry purée into the rest of the yogurt mixture until pale pink. Pour one-third of this on top of the apple layer, then freeze until firm. Repeat twice, adding enough fruit to the yogurt mixture to make each layer slightly darker pink than the last. Freeze overnight or until totally solid. Can be made 1 month ahead, wrapped well in the freezer.
When you’re ready to serve, defrost the remaining blackberry purée, if frozen. Bring the purée, 1⁄4 tsp cinnamon and 1 tbsp of the sugar to a simmer. Stir in the arrowroot mix until slightly thickened and shiny. (Arrowroot does not freeze well, so don’t freeze after this point.) Cool.
Take the cake from the freezer 30 mins before serving. Remove from the tin after 5 mins, peel away the paper and transfer to a plate. Whip the remaining 300ml cream, 250g yogurt and 3 tbsp sugar together until billowy but not stiff. Spoon on top of the cake in peaks and scatter with a few more berries. Add a drizzle of the sauce just before serving, and pass the rest around in a jug.