Preheat the oven to 180 degrees Celsius or 350 degrees Fahrenheit. Pour all of the ingredients, except the eggs and dulce de leche, into a saucepan and bring to the boil.
Reduce the heat and simmer for 15 minutes, stirring constantly, until thickened. Take the pan off the heat and remove the vanilla pod and strips of lemon zest.
Leave to cool for 10 minutes, remove the vanilla pod and lemon zest then beat in the eggs. Meanwhile grease a 20 centimetre sandwich tin with the butter, spread the dulce de leche over the base in an even layer then chill in the fridge until needed.
Pour the cooled rice mixture over the top and even out the top with the back of a spoon.
Place on a baking tray and bake for 25 to 30 minutes until just set, with a slight wobble in the centre. Leave the cake to cool completely then transfer to the fridge and chill until ready to serve.
Just before serving, prepare the apples. Core the apples, cut into wedges and toss with the sugar and cinnamon. Heat the butter in a non-stick frying pan and when foaming add the apples.
Cook for 2 to 3 minutes, shaking the pan occasionally, until then apples are golden brown but still hold their shape. Add the raisins and rum and cook until most of the liquid has evaporated.
Remove from the heat and leave to cool slightly. Turn the rice pudding cake out onto a serving plate or board. Cut into thick slices and serve with the warm apples.