Bring the broth to a full boil in a soup kettle. Gradually add the rice, stirring constantly until the broth boils again. Reduce the heat, cover, and simmer until the rice is just tender, not mushy, 12 to 14 minutes. Remove from the heat and keep warm while preparing avgolemono. Beat the eggs for 2 minutes. Continue to beat, gradually add the lemon juice. Slowly add some of the hot broth to the egg-lemon mixture, beating steadily. Stir the mixture into the soup and cook over minimum heat, without boiling, until the soup thickens to coat a spoon. Taste for salt, and keep warm over hot water until ready to serve. Pass the pepper mill at the table for additional zest. (If desired, bite-sized pieces of cooked chicken may be added to soup before adding the rice).