In a large bowl, stir the rice and pesto together until blended. Divide the rice into 16 more-or-less-equal portions.
Cut off about ½ teaspoon of mozzarella, and then, with your hands, ball up 1 serving of rice around the cheese so it’s completely encased in rice. Gently place on the prepared platter. Repeat to form 16 arancini.
Slide the platter into the freezer for 30 minutes to allow the balls to firm.
Before you take the rice balls from the freezer, set up your dredging station. Pour the flour into a shallow bowl, the eggs into another shallow bowl and the panko into a third shallow bowl.
In a large, heavy pot or Dutch oven, heat 3 inches of oil over medium-high heat until it registers 375 degrees on a deep-fat thermometer.
While the oil heats, dredge each rice ball in flour and lightly shake off the excess.
Dip each rice ball in the egg and then in the panko. Gently drop 4 to 6 balls into the oil and cook until lightly browned, 60 to 90 seconds. Don’t overcook or the cheese will leak out into the oil.
Using a slotted spoon or wire skimmer, transfer the arancini to paper towels to drain. Repeat to cook the remaining arancini. Serve at once.