In a medium-sized bowl, scatter the cooked sushi rice across the bottom in an even layer. Sprinkle the all-purpose flour and kosher salt over the rice. Using a rice paddle or damp hands, gently fold and mix the ingredients together until each grain is lightly coated and no visible flour remains.

Prepare a small bowl of water to help manage stickiness. Divide the rice mixture evenly into four portions. Working one portion at a time, lightly wet your hands and shape the rice into a compact ball. Flatten the ball into a disk approximately 3 inches wide and 1/2 inch thick. Press and smooth the edges to maintain a clean, round shape. Transfer each finished rice cake to a plate as you form them.

Heat 2 tablespoons of vegetable oil in a medium nonstick skillet over medium heat until the oil shimmers and easily coats the pan when swirled. Carefully add the rice cakes to the skillet. Cook undisturbed until the bottoms are a rich golden-brown color, about 5 minutes.

Using a spatula, gently flip each rice cake. Add the remaining tablespoon of oil to the pan. Continue cooking, occasionally sliding the rice cakes around to ensure they soak up the oil and brown evenly, about 4 to 5 minutes longer. Once both sides are deeply golden and crisp, remove from the skillet.

Serve the rice cakes hot, generously dusted with furikake or alongside a small dish of soy dipping sauce for an extra savory touch.