Melt the butter in a large saucepan over medium heat. Add the onions and cook until translucent. Add the rice and stir until you smell a toasty aroma.
Add the wine and bring to a boil. Stir until the rice absorbs the wine.
Lower the heat to medium-low. Add 1/2 cup water with 1/2 tsp. of salt dissolved in it. Stir lightly until the rice has absorbed it all. Repeat this process twice. Then stir in just enough water that the rice is becoming sticky but the grains are not disintegrating. This differs from rice to rice, but the typical amount of water total is about two cups.
Dump the rice onto a metal pan and spread it out about a half-inch deep. Put the pan into the refrigerator to cool the rice.
When the rice is cool, wet one hand and spoon about 2 Tbs. of rice into it, spreading it out in your palm. Pick up about a teaspoon of the grated cheese and put it in the center of the rice. Close your hand to enclose the cheese into the rice. Wet the other hand and roll the rice into a ball.
Roll the rice ball in the flour to coat lightly. Shake off excess. Dip the ball into the beaten egg, and shake off that excess. Finally, roll the ball through the bread crumbs. Repeat until you run out of rice. Place the finished balls on a pan and, when finished, refrigerate. You can do the recipe up to this point in advance.
To cook the rice balls, heat the oil to 350 degrees. Drop the rice balls in, four to six at a time. Roll them around to brown evenly to a medium brown crispness--about three minutes of frying. Drain in a fine sieve over a pan.
Mix all the sauce ingredients and serve at cool room temperature with the rice balls.