Make meatballs: In a large bowl, combine egg, panko, milk, garlic, lemon zest, mint, parsley, salt, and pepper and let sit a few minutes for crumbs to soften. Add lamb, breaking up with your fingers, and mix well with your hands.
With wet hands, form meat mixture into 3/4-in. balls (use about 3/4 tsp. for each) and set on a rimmed baking sheet. Chill until a little firmer, about 15 minutes.
Meanwhile, make soup: In a large covered pot, bring broth and zest to a simmer. Add rice and simmer over low heat, partially covered, until rice is al dente, about 20 minutes.
While soup simmers, heat oil in a large, heavy frying pan (preferably nonstick) over medium-high heat. Cook meatballs in 2 batches, turning once, to brown on 2 sides, about 6 minutes total. Shake pan, rolling meatballs around to brown a little further. With a slotted spoon, transfer meatballs to paper towels.
With a slotted spoon, remove zest from soup and discard. Add meatballs and bring to a simmer over high heat. Turn off heat.
In a medium bowl, whisk eggs to blend and add lemon juice. Whisk in a ladle of hot soup. One at a time, whisk in about 5 more ladles of soup, then return mixture to pot. Rewarm over low heat if needed. Serve immediately with more mint and parsley.
Mastering meatballs. They sound rustic, but they're delicate things that suffer from over-handling. Here are tips for making them come out right.
Mixing: Always start by mixing the bread crumbs (which keep the meatballs from getting tough), any liquids, and seasonings—then add the meat. Stop when the mixture looks evenly blended; don't overwork it.
Forming: Dampen your hands with cold water to keep the meat mixture from sticking while you roll it into balls. If it's still sticky, let it sit for a few minutes in the fridge.
Browning: A good crust helps meatballs retain their shape—important if they're going to simmer in a thick sauce. If you plan to eat them on their own or add them to broth, though, a light pan-frying will do the trick.
Note: Nutritional analysis is per 1 1/2-cup serving.