Make the broth. Into a large soup pot or Dutch oven, add the water, two tablespoons of olive oil, and one teaspoon of salt. Bring to a boil over high heat, then reduce the heat to medium-low to allow the water to simmer while you roll the meatballs.

Make the meatballs. In a large bowl, combine the beef, rice, onion, egg, parsley, mint, salt, pepper, and olive oil. Mix the ingredients well, then roll approximately 40 small meatballs with your palms (they should each be a little smaller than a golf ball).

Cook the meatballs. Carefully transfer the meatballs to the simmering water, you may need to push the meatballs aside with a spoon as you go to make room. Adjust the heat to maintain a simmer. Cook until the meatballs are cooked through, about 30 minutes.

Make the Avgolemono. While the meatballs cook, separate the eggs, placing the whites in a medium bowl and the yolks in a small bowl. Whisk the whites until they are thick and fluffy but not too stiff. Add the yolks and lemon juice to the egg whites and whisk until fully incorporated. It will look very thick and yellow, whisk vigorously to fully incorporate.

Temper the eggs. When the meatballs have cooked through, take one large ladle of broth from the pot of meatballs and slowly drizzle the liquid into the egg and lemon mixture while whisking continuously. Repeat with a second ladleful of broth. Pour the egg and lemon mixture into the soup, while stirring. At this point, it should be foamy, still watery, and light yellow in color. Turn the burner off, cover the pot with a lid and let the soup sit for about 10 minutes to thicken.

Serve. Divide the soup among bowls and finish with freshly cracked black pepper. Serve with lemon wedges and, optionally, bread on the side.