For the rice cakes: In a bowl, combine the rice, breadcrumbs, egg, and season with salt and pepper. Mix together until well until incorporated. Take 1/4 cup of the rice mixture and form into 1/2-inch thick patty. Set aside while you form the other 3 patties.
In a nonstick skillet over medium heat, add the oil. When it is hot, fry the rice cakes in batches until they are golden brown and crispy, 3 to 4 minutes per side. Drain on a brown paper bag.
For the creamy cilantro sauce: In the bowl of a food processor, add the cilantro, vinegar, garlic, and sour cream and process to combine. Taste and adjust the seasoning with salt and pepper.
Serve the rice cakes with a dollop of the Creamy Cilantro Sauce.