Preheat the oven to 190C/375F/Gas 5. Grease and flour a 23cm/9in spring-form tin.
Melt 25g/1oz of the butter in a saucepan and add the apples, water and sugar. Cook over a gentle heat until steam appears from the saucepan, then cover with a lid and cook for 3-4 minutes, or until thick and fluffy.
Remove the lid and beat the mixture to remove any lumps. Add the calvados and beat until well combined.
Beat 90g/3oz of the butter and the soft brown sugar in a large mixing bowl until light and fluffy.
Add the golden syrup, eggs and vanilla extract and mix until well combined.
Fold the self-raising flour into the cake mixture.
Meanwhile, add the bicarbonate of soda to the apple puree and mix well, then stir this quickly into the cake mixture. Pour into the cake tin and gently tap the sides of the tin to evenly disperse the mixture.
Bake for about 40-45 minutes in total. While the cake is baking, melt the remaining 25g/1oz of butter in a saucepan. After the cake has been in the oven for 30 minutes, arrange the sliced apple over the top of the cake in a circle, and brush with the melted butter. Sprinkle brown sugar evenly over the cake and return to the oven. Once the cake is cooled, allow to cool slightly before turning out.
For the calvados caramel sauce, place the sugar and butter into a small saucepan and cook until melted and well combined. Pour in the double cream and calvados. Simmer gently for three to five minutes, or until the mixture thickens slightly.
To serve, cut the cake into slices and put a spoonful of ice cream on top. Finish by drizzling over some of the sauce.