Combine water, rice and salt in a saucepan. Bring to the boil, then cover and lower heat. Simmer until rice is tender, about 10 to 15 minutes. Drain any excess water.
Over medium heat, add 450ml of the milk to the rice, along with the sugar and vanilla. Simmer, stirring occasionally, for about 10 minutes. Remove from heat.
Beat the egg yolks in a large mixing bowl. Slowly ladle in some of the hot milk from the saucepan, whisking vigorously so that the egg yolks do not curdle. After whisking in 1 or 2 ladle-fuls of the hot milk, pour the egg mixture into saucepan with the remaining rice-milk mixture, whisking well. Stir in the chopped apples.
Preheat the oven to 180 C / Gas 5. Grease a 20cm square baking dish or Pyrex.
Tip the mixture into the prepared dish. Place the dish in a roasting tin lined with a damp tea towel. Fill the roasting tin with hot water so that it comes halfway up the sides of the baking dish.
Bake in the oven for about 30 minutes, or until the pudding is just set in the centre. Allow to cool slightly, and serve warm or cold, dusted with cinnamon if desired.
You can save on fat and calories by using semi-skimmed milk in this pudding.