In a small bowl, combine yogurt with 2 Tablespoons water. Whisk until smooth. Set aside. If you don't have yogurt, use a 1/4 cup milk instead. I don't keep fresh milk on hand usually, so this yogurt/water mixture is my solution (I almost always have yogurt!).
For the Meatballs
Put ground lamb in a medium-sized bowl with yogurt mixture, egg, breadcrumbs, salt, and pepper.
Grate a clove of garlic over the bowl with a microplane. Alternatively, finely mince and add to the bowl.
Zest 1/2 lemon over the bowl. Mince about 1 teaspoon EACH parsley and mint and add to the bowl. Combine mixture well with your hands, breaking up lamb with your fingers.
With wet hands, form mini meatballs (they should be bite-sized) and set on a rimmed baking sheet. You should have around 40 meatballs. Refrigerate for 15 minutes.
For the Soup
Make wide ribbons of zest from 1/2 lemon, using a vegetable peeler. Place in a large pot with a lid with broth. Cover and bring to a simmer.
Add rice and simmer over low heat, partially covered, until rice is just cooked, about 20 minutes.
Meanwhile, heat a large heavy frying pan over medium-high heat. Add olive oil and swirl around the pan. Add meatballs (you'll need to do this in 2 batches) and cook until brown on 2 sides, turning once. Remove meatballs to a plate lined with paper towels or a paper bag.
Remove the ribbons of zest from the soup and discard. Add the meatballs, bring to a simmer over high heat, and turn off the heat.
Whisk the eggs in a medium bowl, and add juice of 1 lemon. Whisk in a ladle of soup, stirring after each addition. Add about 5 more ladles of soup. Let large pot of soup cool for 2-3 minutes. Return egg/broth mixture to the pot, and serve immediately, garnished with mint and parsley.
*ground turkey or chicken can be used as a substitute