Combine the ground beef, rice, onion, garlic, parsley, mint, salt, and pepper in a bowl. Add an egg and mix well.
Pour some flour in a shallow pan. With floured hands, shape the meat into walnut-sized balls. Lightly roll the meatballs in the flour and shake off the excess.
Heat 2 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. If the broth does not cover the meatballs, add a bit more broth until they are covered.
Simmer, covered, on low for 45 minutes, adding more broth if there is less than 1 cup remaining.
To make the sauce, beat the eggs in a medium sized bowl until frothy.
Slowly whisk in the lemon juice. Ladle one cup of the pot liquid, little by little, into the egg mixture to temper the eggs.
Remove the pot from the heat and add in the lemon/egg mixture, stirring gently.
Heat over very low heat until sauce thickens and is heated through, making sure the sauce does not boil (which will result in the eggs curdling).