Step 1: Mix the Chicken Meatball Base - Add the ground chicken, shredded zucchini, breadcrumbs, Parmesan, about 1/4 cup of the mozzarella, salt, one minced garlic clove, and the egg to a large bowl. I like to use my hands for this because you can feel when everything has been distributed evenly, but try not to squeeze and mash the mixture any longer than necessary. You want the ingredients combined rather than packed down. Before adding the zucchini, give it a gentle squeeze or let it drain briefly so you are not adding a puddle of extra liquid to the bowl.
Step 2: Shape the Meatballs - Wet your hands with a little cool water before you start shaping. Ground chicken is naturally sticky, and damp hands make this job much less frustrating. If you have a number 20 scoop, use it to portion the mixture so the meatballs are close to the same size. If you do not have one, a regular spoon works just fine. The important thing is consistency because meatballs that are similar in size will cook at roughly the same rate. You should end up with about 20 meatballs. Set them on a clean plate as you work.
Step 3: Brown the Meatballs - Heat 2 tablespoons of olive oil in a large skillet over medium heat. Once the oil is warm, carefully add the meatballs in a single layer. Give them some breathing room so they brown instead of steaming. Let the first side develop a golden surface before carefully turning each meatball to brown the other side. They do not need to be completely cooked at this stage. What you are after is color and that extra savory flavor that develops where the chicken meets the hot skillet. Transfer the browned meatballs to a plate.
Step 4: Start the Tomato Sauce - Keep the same skillet on the stove and add the remaining tablespoon of olive oil. Add the second minced garlic clove and stir it briefly, just until the garlic smells fragrant. Pour in the tomato sauce and stir, scraping gently across the bottom of the skillet so the browned bits left behind by the meatballs make their way into the sauce. I love this part because it means all the good flavor you built during browning stays right in dinner instead of getting washed down the sink.
Step 5: Simmer the Meatballs in the Sauce - Return the meatballs to the skillet and nestle them into the tomato sauce. Cover the skillet and let them cook for about 10 minutes, then carefully turn them so they cook evenly. Continue cooking until the chicken meatballs are completely cooked through and tender in the centers. Keep an eye on the consistency of the sauce while they simmer. If it gets thicker than you would like, stir in a small splash of water. You want enough sauce left to spoon over the meatballs and whatever pasta or rice you serve alongside them.
Step 6: Melt the Mozzarella - Sprinkle the remaining mozzarella evenly over the meatballs. Do not worry about covering every inch with mathematical precision. Some cheese will land directly on the meatballs and some will melt into the tomato sauce, which is exactly what you want. Scatter a little fresh basil over the top, cover the skillet again, and cook for about 5 minutes. When you lift the lid, the mozzarella should be soft, glossy, and melted across the skillet.
Step 7: Finish With Fresh Basil and Serve - Take the skillet off the heat and add another handful of fresh basil leaves. I like saving some basil for this final moment because the fresh leaves bring a brighter aroma than basil that has cooked in the sauce. Serve the meatballs hot, making sure every plate gets plenty of tomato sauce and melted mozzarella. Add pasta, rice, roasted vegetables, or a piece of bread on the side and dinner is ready.