For the egg-lemon sauce
In a large bowl, mix together the minced beef, onion, rice, dill, mint, 3 tablespoons of olive oil, salt and pepper, using your hands. Knead the mixture for a few minutes, and shape small round "giouvarlakia" (medium meatballs size). Place them next to each other.
In a large saucepan, boil the meat broth and the remaining olive oil. Pour the mixture over the giouvarlakia and cook at low temperature for about 50 minutes.
In a bowl, beat the eggs with a fork. Add the lemon juice and while you continue beating, add a few tablespoons of the soup broth. Pour the egg-lemon sauce (avgolemono in Greek) over the giouvarlakia. Shake the saucepan and cook for another 1-2 minutes.