In a skillet on medium heat cook the onion in butter until translucent, add the mushrooms and cook until mushrooms are tender then set aside.
In a saucepan place the chicken broth, olive oil, salt and bring to a boil. Add the rice and the saffron and stir. Cover and turn heat to low-medium. Cook for about 20 minutes stirring every 5 minutes. Make sure the broth is absorbed and that the rice is "al dente".
Transfer rice to a bowl and allow to cool. When rice is cool to the touch, add the Parmesan/Pecorino cheese and mix well. Add 1 beaten egg and mix again.
Scoop rice using a ½ cup measure and flatten slightly on the palm of your hand. Place a tablespoon of the mushroom mixture in the middle of the rice. Scoop another ½ cup of rice and place on top and roll to form into a ball.
Prepare two pie plates: one will contain the remaining two beaten eggs, the other will have breadcrumbs.
Dip each ball into the egg mixture to coat and roll in the breadcrumbs. Place prepared rice balls on a tray or cookie sheet and place in the fridge for about 30 minutes to allow the breadcrumbs to set properly.
Heat oil to about 375°F. If you don't have a thermometer, you can test the temperature by lightly dipping a rice ball into the oil. If it begins to sizzle, it is ready for frying.
Fry a few at a time. Arancini should be golden brown, not dark brown.
Drain on paper towel.
Serve promptly.
You can serve your arancini with or without marinara sauce.