Sara: "What...is that?"
Me: "What are rice cakes?"
Jackie: "You mean those hard things you toast and slather peanut butter on?"
Renne: "Or do you mean something like arancini?"
Me: "My mom takes leftover rice, mixes it with an egg and a little bit of flour, and fries it into little rice fritters that she dumps a lot of salt on. This isn't something that other people's moms do? I assumed it was like a regular mom dish."
In discussing it with other friend groups this week, it turns out that nobody's mom made these. I was asked a few times if I meant arancini, but these didn't have a breadcrumb coating, and they weren't round. They were like little patties, crispy on the edges and soft in the middle, and each time I explained them nobody seemed to know what I was talking about.
I was surprised by this, because my mom's rice cakes are the meal I remember best from childhood.
My mom is a good cook, but my mom is also a home cook, which means that a lot of her recipes involve pinches, dashes, and just knowing by looking if something is the right consistency or if it needs a little more flour. She makes lots of things well, and many nights she made two dinners because I was such a picky eater, so I remember getting a lot of things that were variations on other things: white pizza, sauceless lasagna, spaghetti with olive oil and garlic instead of red sauce, etc. I remember the rice cakes as a treat, though, something we usually only got on weekends and not every weekend. In my head, my mom made them out of leftover rice from dinner, but I think she actually made rice just to make these.
Last time I attempted to make these was in 2007, and they did not come out as well as I remember Mom's. In retrospect, I overanalyzed Mom's recipe, and tried to make it exact when it's not really an exact sort of process. Ten years later, I am a better, more confident cook, so I decided to ask Mom for the recipe again and give these another try.
Now that I have, here's an updated recipe, which I have been given permission to share:
2 cups of cooked rice (however much one cup of uncooked rice makes)
2 eggs, beaten
3 tablespoons of flour
a pinch of sugar
a pinch of salt
Leave the salt container out, because you're going to need it again.
1) Put your rice in a bowl.

If you have a Pyrex bowl of any sort, that's best, but if you don't then just use whatever mixing bowl you have.
2) Mix everything together.

I mixed the eggs, sugar, and salt in first, then mixed the flour in after.
3) Heat 1/8-1/4 of an inch of oil in a frying pan with decently high sides. The oil is going to spit a little, and you don't want it all over the stovetop. I set the heat exactly between medium and high. You want these to get crispy on the edges, but not so hot that they brown too fast.
4) Spoon the rice mixture into the oil and use the back of the spoon to flatten it. It should look like a little patty.

5) When the edges look golden brown, turn it over and let it cook for about another minute. (It won't take as long as the first side because that side partially cooked while the other side was cooking, like pancakes do.)
6) Using a slotted spatula, remove it from the oil, let it drain for a few seconds, and then put it on a plate that's covered in paper towels and immediately sprinkle salt on top.

7) Now that you've built up your confidence by doing it right one time, start making more than one at once.

It makes about this many rice cake, plus two that I ate while I was making the others:

If you're trying to cut salt out of your diet, try adding something to the batter to flavor these, like chives or a spice blend.
Also, eat them while they're still warm, or room temperature. If you put them in the fridge to get cold, they get kind of gummy, and I have no idea how to reheat these.