Bring the water or stock, olive oil and salt to a simmer in a large wide pot.
While the liquid is simmering, prepare the meat-rice balls: In a large bowl, combine the ground beef or combination ground beef and pork, rice, onion, egg, parsley, mint, salt, pepper, and olive oil. Knead the ingredients well, then roll the meatballs using the palms of your hands to make approximately 40 to balls a little larger than a golf ball.
Using a large spoon or slotted spoon, carefully slide the meatballs into the simmering liquid. Adjust the heat to maintain a simmer. Cook with the lid ajar until the meatballs are done and the rice tender, about 30 to 35 minutes.
While the meatballs simmer, separate the eggs. Whisk the whites to form a fairly stiff meringue. In a separate bowl, whisk the yolks until they are thick and frothy. Slowly whisk in the lemon juice to the egg yolks and then add this, slowly, whisking all the while, to the whipped whites.
When the meatballs have cooked through, turn off the heat. Take a large ladle of hot broth from the pot and very slowly drizzle the liquid into the egg and lemon mixture while whisking continuously and vigorously. Repeat with a second ladleful of broth. Pour the egg and lemon mixture into the soup, and tilt the pot so that it is evenly distributed. Cover the pot with a lid for 3 to 5 minutes to let the soup thicken. Don’t leave the lid on too much longer because you’ll end up with an omelet!
Serve immediately.