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Heat 1 tbsp butter and 1 tbsp sugar in a large frying pan over medium heat until the sugar starts to melt. Add apples, then cook for about 5 mins, until nicely browned and the sauce is a rich caramel. Leave to cool.
While the apples cool, make the topping. In a food processor or using the tips of your fingers, rub together flour, 85g sugar, cinnamon and cold butter until the mixture is crumbly and only pea-size pieces of butter remain. Stir in 2 tbsp of hazelnuts. Set aside in a cool place.
Heat oven to 180C/160C fan/gas 4. To make the cake, place the remaining butter and sugar in a bowl or food mixer and beat until light and creamy. Add the eggs one at a time, followed by the vanilla extract. Add the flour, baking powder, cinnamon and milk in two goes, alternating between dry and wet ingredients. When the mixture is well combined, stir through the apples.
Grease a 20cm cake tin with removable base and line the base with a circle of baking paper. Spoon in the cake mixture, smooth over the top with a spoon, then sprinkle over the streusel topping. Scatter with the remaining 1 tbsp demerara sugar and hazelnuts. Cook in the oven for 50 mins-1 hr or until a skewer inserted into the cake comes out clean. Leave to cool in the tin for 10 mins, then turn out onto a baking rack, remove paper and cool.