Combine rice and water in a saucepan, bring to a gentle simmer, cover, and cook until tender and creamy, about 15 minutes.

While rice cooks, set your oven to 350°F (175°C) to preheat.

In another pan, gently heat milk, heavy cream, and sugar over medium-low heat until steam rises and sugar dissolves.

Crack eggs into a bowl, whisk until smooth, then slowly stream in a bit of warm milk mixture to temper, whisking constantly.

Stir tempered eggs back into the remaining milk mixture, fold in cooked rice, and pour into a buttered baking dish.

Place the dish on a middle rack, bake at 350°F for about 35–40 minutes until set with a slight jiggle in the center.

Remove from oven and let it cool for at least 10 minutes before serving.