Both Meme and Grandma Marian were of the school of thought that rice should be cooked forever in the top of a double boiler. I know Meme's mother (my great-great-grandmother) cooked it in the same manner, and it was always served on Saturday night for supper. Thus, I almost laughed aloud when I read the following bit from page 50 of my reprint of the 1933 Home Comfort Cookbook:
"Rice should not be put into a double boiler and cooked slowly into a pasty mass."
That is EXACTLY what they both did, and I ate it happily every time!
If I had to hazard a guess, I would say that Grandma only made rice once per month. It was served instead of the ever-present potato with a sprinkling of sugar and a dash of cinnamon once it was on our plates. Other than that, it was completely unseasoned. (I had never even heard of salting rice until I was in college, and I had never had Chinese food either.) If any rice was leftover, she would make it into rice pudding for dessert a day or two later. Since she didn't ever know how much rice she would have (she was constantly cooking for large groups of family and hired men), there was never any recipe for her rice pudding. Furthermore, sometimes it all fit into a small custard cup, and sometimes it needed an 8" x 10" Pyrex dish. Obviously, this dish is very flexible.
Also, I should give fair warning that though Grandma always called this "rice pudding," it is really more of a baked custard with rice in it. Thus, if you are looking for a rice pudding that is more similar to a cornstarch pudding, this isn't going to fit that bill.
To start with, you need leftover cooked rice, eggs, milk, sugar, and vanilla. You can see in the picture below that I had two different containers of leftover rice, and I would estimate that they amounted to about 2 1/2 cups altogether.
We have some family friends who make a very similar dish that they just call "Baked Rice." Theirs is less sweet, has a bit of salt added to it, and is served as a starchy side dish for the main meal rather than a dessert, but it is the same basic idea. Both their family and Grandma's has a lot of German influence, so I wonder if this is a German dish.
Please fill up the comments with information about whether you or your family make something similar, and let us all know how your method differs.