Place 12 ounces ground meat, ½ cup dry breadcrumbs, 1 of the eggs, ¼ cup of the Pecorino, ¼ cup of the Parmesan, oregano, 1 teaspoon kosher salt, and ½ teaspoon black pepper in a bowl. Mix thoroughly but be careful not to overwork the meat. Form the mixture into ¾-inch to 1 ½-inch balls. You should have 20 to 30 meatballs, depending on how large you form them.

Heat 2 tablespoons of the oil in a large skillet over medium-high heat until shimmering. Add the meatballs in batches so as not to crowd the pan and cook, turning occasionally, until browned all over, 3 to 5 minutes. (If they are still a bit pink in the middle, don't worry; they will continue to cook in the broth.) Transfer to a paper towel-lined plate as each batch is ready.

Heat the remaining 1 tablespoon oil in a 4- to 6-quart soup pot over medium-high heat. Add the diced yellow onion and 4 cloves minced garlic and sauté until the onions are tender and the garlic is soft but not browned, about 5 minutes. Add 8 cups low-sodium chicken broth and bring to a boil. Add 6 (lightly packed) cups of torn greens, reduce the heat to low, cover, and simmer for 10 minutes. Add the meatballs and cook 5 minutes more. Meanwhile, combine the remaining 2 eggs, remaining ¼ cup Pecorino, and remaining ¼ cup Parmesan in a small bowl and beat with a fork to blend.

Slowly pour the egg mixture into the simmering soup, stirring slowly in one direction. Cover and simmer just until egg bits are set, about 30 seconds. Stir in ¾ cup cooked pasta, if using. Taste and season with kosher salt, black pepper, red pepper flakes, and a sqeeeze of lemon juice as desired. Serve immediately.