These are a breakfast staple of my (and now my kids') childhood. We call them "rice cakes" though "rice pancakes" would be a more accurate description. I don't know where the recipe originates, but it goes back to my great grandmother at least, and maybe further. Her line came to Indianapolis from Germany in the mid 1800s. No one else (outside my extended family) seems to have heard of these.
It's kind of an "eyeball it" recipe, because it starts with last night's leftover white rice, of which you might have a varying amount. Historically this was always home cooked rice, but now I sometimes use leftover rice from Chinese or Thai takeout which works just as well.
1) Put leftover rice in a mixing bowl
2) Add 1 or 2 eggs. (2 eggs for a full pint-sized container of takeout rice)
3) Add milk until it starts to rise just above the rice around the edges of the bowl.
4) Add two heaping dinner spoons of flour
5) If your rice doesn't have salt in it already, add 1 tsp (ish) of salt
6) Mix thoroughly by hand
I use a 1/3 cup measure to ladle out onto a griddle. The "batter" should be significantly runnier than pancake batter — a bit like crepe batter — and make kind of a flange around the rice "patty" on the griddle. Flip them when the flange starts to look relatively firm, and be careful — they're a bit more fragile than pancakes.
We eat them with butter and syrup just like pancakes.