Pour oil into a large, wide, heavy pot to a depth of 3 in. and heat until oil measures 350° on a deep-fry thermometer.
In a glass measuring cup, beat egg; pour half into a large bowl (save the rest for another use). Add chilled risotto and green onions to egg in large bowl and stir until well combined.
Cut fontina into 1/2-in. cubes. With wet hands, form 2 tbsp. risotto mixture into a ball and make a well in the center with your thumb. Put a cube of fontina in the well and push risotto mixture up around it to form a ball. Set ball on a baking sheet and repeat with remaining risotto mixture and fontina.
Put bread crumbs in a shallow dish. Roll balls in crumbs; return to baking sheet. Line another baking sheet with paper towels.
Fry first ball as a test: Cook, turning, until golden all over, 2 to 4 minutes. Drain on lined baking sheet and cut to see whether cheese has melted. If it hasn't, raise temperature a little or cook a little longer. Fry balls in batches of 5 or 6, turning, until golden all over, 2 to 4 minutes per batch. Using a slotted spoon, transfer as done to lined baking sheet. Season with salt and serve immediately, or keep warm in a 200° oven up to 30 minutes.
Note: Nutritional analysis is per risotto ball.