Fill a saucepan halfway with vegetable oil and heat until a deep-fry thermometer inserted in the oil registers 350 degrees F.
Put some flour in a shallow bowl and season with salt and pepper. Put the eggs and panko in separate shallow bowls.
Form the rice into 1 1/2-inch balls. Push some of the cheese and ham into the center of each ball and press the rice around the filling. Dredge the rice balls in the seasoned flour, shaking off the excess, then in the egg followed by the panko.
Place a few of the balls in the hot oil and cook until golden brown all over, about 3 minutes. Repeat with the remaining rice balls. Drain slightly on paper towels and serve.
This is a great opportunity to make use of overcooked or not-so-perfect rice.