Pour the hot vegetable stock and the water into the slow cooker, and add the butter and 1/4 teaspoon of the salt.
In a large mixing bowl, combine the beef, rice, onion, parsley, tarragon, oil, the remaining 1 1/2 teaspoons (9 g) of salt and the pepper. Mix well with your hands, kneading lightly for a couple of minutes.
Take three level tablespoons of the mixture and shape it into a large meatball. Set aside on a plate. Continue with the rest of the mixture and then gently drop all the meatballs into the slow cooker. There should be about 22 balls.
Cook on low for 5 to 6 hours, or until the meatballs are cooked through. Turn the slow cooker off.
In a medium-sized mixing bowl, beat the eggs and lemon juice until well combined. Place the bowl next to the slow cooker, and use a ladle or measuring cup to slowly add small amounts of hot soup into the egg mixture, beating continuously. Start with about 1 tablespoon (15 ml) at a time, and gradually increase the amount of soup added to the bowl. When at least 2 to 3 cups (480 to 720 ml) of soup have been transferred to the egg mixture, and the bowl feels hot to the touch, pour the mixture back into the insert while stirring gently.
Serve the soup with freshly ground black pepper, fresh crusty bread and feta cheese.