Combine ground beef, rice, onion, garlic, parsley, mint, salt, and pepper in a bowl. Add egg to mixture and mix well. Place flour in a shallow pan. With floured hands, shape into walnut-sized balls. Roll meatballs lightly in flour and shake off excess.
Heat two cups of the chicken broth in a Dutch oven or large pot until boiling. Carefully place the meatballs in a layer on the bottom of the pot. Add more chicken broth if needed, just to cover the meatballs. Simmer covered, over low heat for 45 minutes. Add more broth if less than one cup remains.
For the Egg-Lemon Sauce: Using a whisk, beat the eggs in a medium bowl until frothy. Slowly whisk in the lemon juice. Ladle one cup of the pot liquid little by little into egg-lemon mixture to temper the eggs. Remove pot from heat and add egg-lemon mixture stirring gently. Heat over very low heat until sauce thickens and is heated through. Take care not to allow the sauce to boil or the eggs will curdle.
Top meatballs with sauce and serve over orzo or rice.