Rinse rice under running water. Grease a 20 cm x 30 cm (8-inch x 12-inch) deep baking dish with butter and set aside until needed. Place a rack in the middle of the oven and preheat the oven to 180 °C / 350 °F.
cook the rice pudding
In a pot, combine rinsed rice, water, milk, and a pinch of salt. Cover with a lid and place over medium-high heat. Bring to a boil, then simmer for 25 - 30 minutes. When the rice cooks, set aside and bring to room temperature.
beat the eggs
Separate the eggs. Add the yolks to a large bowl, and add the whites to a smaller bowl. Add sugar, lemon zest, and vanilla to the yolks and beat for 2 - 3 minutes on high speed to get a smooth mixture. Add the softened butter and beat for another 2 minutes to combine. Whip the egg whites separately with a clean whisk until stiff peaks form.
combine all the ingredients
Add room-temperature rice pudding to the egg yolk mixture. Stir with a spatula to combine, then gradually gently fold in the egg whites. Try to keep as much volume as possible.
Pour the rice pudding mixture into the prepared baking dish. Place in the preheated oven on the middle rack. Bake for 40 - 45 minutes at 180 °C / 350 °F or until golden brown. Keep an eye on the rice pudding for the last 15 minutes to prevent burning. If you feel like the rice pudding is getting too brown too quickly, cover it with aluminum foil (make sure the foil doesn't touch the food) and continue baking.
When the rice pudding bakes, cut it into eight slices. Optionally dust with icing sugar and serve.