Rinse the rice in a fine-mesh strainer under cold water until it runs clear. In a small saucepan, combine the rice and 1/2 cup (120ml) water. Bring to a boil, then reduce heat to low, cover, and cook for 10-12 minutes until the water is absorbed. Set aside to cool.
While the rice cooks, prepare the meatball mixture. In a large bowl, combine the ground beef, cooled rice, beaten egg, breadcrumbs, 1 teaspoon of kosher salt, and the black pepper. Mix gently with your hands until just combined.
Dampen your hands with water and form the mixture into 1- to 1.5-inch meatballs, making about 12 to 16. Place them on a plate and set aside.
Heat the olive oil in a large pot or Dutch oven over medium heat. Add the onion, carrot, and celery. Sauté for 5-7 minutes, until the vegetables begin to soften.
Pour in the chicken broth and increase the heat to medium-high. Add the diced potatoes, diced tomatoes, the remaining 1/2 teaspoon of kosher salt, and the oregano sprig.
Bring the broth to a simmer. Carefully lower the meatballs into the soup one by one. Reduce the heat to maintain a gentle simmer.
Cook for about 20-25 minutes, or until the potatoes are tender and the meatballs are cooked through. The meatballs should float to the surface. Confirm they are done by checking that the internal temperature reaches 160°F (71°C) with a meat thermometer.
Remove the oregano sprig if you used a fresh one. Taste the broth and adjust seasoning with more salt if needed. Serve hot with avocado, cilantro, and lime wedges.